Combustion

Combustion

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Combustion Fourth Edition

Irvin Glassman Richard A. Yetter

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

Academic Press is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK Copyright © 2008, Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (44) 1865 843830, fax: (44) 1865 853333, E-mail: [email protected]. You may also complete your request online via the Elsevier homepage (http://elsevier.com), by selecting “Support & Contact” then “Copyright and Permission” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN: 978-0-12-088573-2

For information on all Academic Press publications visit our Web site at www.elsevierdirect.com Typeset by Charon Tec Ltd., A Macmillan Company. (www.macmillansolutions.com) Printed in the United States of America 08 09 10 9 8 7 6 5 4 3

2

1

This Fourth Edition is dedicated to the graduate students, post docs, visiting academicians, undergraduates, and the research and technical staff who contributed so much to the atmosphere for learning and the technical contributions that emanated from Princeton’s Combustion Research Laboratory.

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No man can reveal to you aught but that which already lies half asleep in the dawning of your knowledge. If he (the teacher) is wise he does not bid you to enter the house of his wisdom, but leads you to the threshold of your own mind. The astronomer may speak to you of his understanding of space, but he cannot give you his understanding. And he who is versed in the science of numbers can tell of the regions of weight and measures, but he cannot conduct you hither. For the vision of one man lends not its wings to another man. Gibran, The Prophet The reward to the educator lies in his pride in his students’ accomplishments. The richness of that reward is the satisfaction in knowing the frontiers of knowledge have been extended. D. F. Othmer

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Contents

Prologue Preface

CHAPTER 1. CHEMICAL THERMODYNAMICS AND FLAME TEMPERATURES

xvii xix

1

A. B. C. D.

Introduction Heats of reaction and formation Free energy and the equilibrium constants Flame temperature calculations 1. Analysis 2. Practical considerations E. Sub- and super sonic combustion thermodynamics 1. Comparisons 2. Stagnation pressure considerations Problems

1 1 8 16 16 22 32 32 33 36

CHAPTER 2. CHEMICAL KINETICS

43

A. Introduction B. Rates of reactions and their temperature dependence 1. The Arrhenius rate expression 2. Transition state and recombination rate theories C. Simultaneous interdependent reactions D. Chain reactions E. Pseudo-first-order reactions and the “fall-off” range F. The partial equilibrium assumption G. Pressure effect in fractional conversion H. Chemical kinetics of large reaction mechanisms 1. Sensitivity analysis 2. Rate of production analysis 3. Coupled thermal and chemical reacting systems 4. Mechanism simplification Problems

43 43 45 47 52 53 57 60 61 62 63 65 66 68 69

x

Contents

CHAPTER 3. EXPLOSIVE AND GENERAL OXIDATIVE CHARACTERISTICS OF FUELS A. B. C. D.

75

Introduction Chain branching reactions and criteria for explosion Explosion limits and oxidation characteristics of hydrogen Explosion limits and oxidation characteristics of carbon monoxide E. Explosion limits and oxidation characteristics of hydrocarbons 1. Organic nomenclature 2. Explosion limits 3. “Low-temperature” hydrocarbon oxidation mechanisms F. The oxidation of aldehydes G. The oxidation of methane 1. Low-temperature mechanism 2. High-temperature mechanism H. The oxidation of higher-order hydrocarbons 1. Aliphatic hydrocarbons 2. Alcohols 3. Aromatic hydrocarbons 4. Supercritical effects Problems

75 75 83 91 98 99 103 106 110 112 112 113 117 117 127 129 139 141

CHAPTER 4. FLAME PHENOMENA IN PREMIXED COMBUSTIBLE GASES

147

A. Introduction B. Laminar flame structure C. The laminar flame speed 1. The theory of Mallard and Le Chatelier 2. The theory of Zeldovich, Frank-Kamenetskii, and Semenov 3. Comprehensive theory and laminar flame structure analysis 4. The laminar flame and the energy equation 5. Flame speed measurements 6. Experimental results: physical and chemical effects D. Stability limits of laminar flames 1. Flammability limits 2. Quenching distance 3. Flame stabilization (low velocity) 4. Stability limits and design E. Flame propagation through stratified combustible mixtures F. Turbulent reacting flows and turbulent flames 1. The rate of reaction in a turbulent field 2. Regimes of turbulent reacting flows 3. The turbulent flame speed

147 151 153 156 161 168 176 176 185 191 192 200 201 207 211 213 216 218 231

Contents

xi

G. Stirred reactor theory H. Flame stabilization in high-velocity streams I. Combustion in small volumes Problems

235 240 250 254

CHAPTER 5. DETONATION

261

A. Introduction 1. Premixed and diffusion flames 2. Explosion, deflagration, and detonation 3. The onset of detonation B. Detonation phenomena C. Hugoniot relations and the hydrodynamic theory of detonations 1. Characterization of the Hugoniot curve and the uniqueness of the C–J point 2. Determination of the speed of sound in the burned gases for conditions above the C–J point 3. Calculation of the detonation velocity D. Comparison of detonation velocity calculations with experimental results E. The ZND structure of detonation waves F. The structure of the cellular detonation front and other detonation phenomena parameters 1. The cellular detonation front 2. The dynamic detonation parameters 3. Detonation limits G. Detonations in nongaseous media Problems

261 261 261 262 264

CHAPTER 6. DIFFUSION FLAMES

311

A. Introduction B. Gaseous fuel jets 1. Appearance 2. Structure 3. Theoretical considerations 4. The Burke–Schumann development 5. Turbulent fuel jets C. Burning of condensed phases 1. General mass burning considerations and the evaporation coefficient 2. Single fuel droplets in quiescent atmospheres D. Burning of droplet clouds E. Burning in convective atmospheres

311 311 312 316 318 322 329 331 332 337 364 365

265 266 276 282 286 293 297 297 301 302 306 307

xii

Contents

1. The stagnant film case 2. The longitudinally burning surface 3. The flowing droplet case 4. Burning rates of plastics: The small B assumption and radiation effects Problems

365 367 369 372 374

CHAPTER 7. IGNITION

379

A. Concepts B. Chain spontaneous ignition C. Thermal spontaneous ignition 1. Semenov approach of thermal ignition 2. Frank-Kamenetskii theory of thermal ignition D. Forced ignition 1. Spark ignition and minimum ignition energy 2. Ignition by adiabatic compression and shock waves E. Other ignition concepts 1. Hypergolicity and pyrophoricity 2. Catalytic ignition Problems

379 382 384 384 389 395 396 401 402 403 406 407

CHAPTER 8. ENVIRONMENTAL COMBUSTION CONSIDERATIONS

409

A. Introduction B. The nature of photochemical smog 1. Primary and secondary pollutants 2. The effect of NOx 3. The effect of SOx C. Formation and reduction of nitrogen oxides 1. The structure of the nitrogen oxides 2. The effect of flame structure 3. Reaction mechanisms of oxides of nitrogen 4. The reduction of NOx D. SOx emissions 1. The product composition and structure of sulfur compounds 2. Oxidative mechanisms of sulfur fuels E. Particulate formation 1. Characteristics of soot 2. Soot formation processes 3. Experimental systems and soot formation 4. Sooting tendencies 5. Detailed structure of sooting flames

409 410 411 411 415 417 418 419 420 436 441 442 444 457 458 459 460 462 474

xiii

Contents

6. Chemical mechanisms of soot formation 7. The influence of physical and chemical parameters on soot formation F. Stratospheric ozone 1. The HOx catalytic cycle 2. The NOx catalytic cycle 3. The ClOx catalytic cycle Problems

478 482 485 486 487 489 491

CHAPTER 9. COMBUSTION OF NONVOLATILE FUELS

495

A. Carbon char, soot, and metal combustion B. Metal combustion thermodynamics 1. The criterion for vapor-phase combustion 2. Thermodynamics of metal–oxygen systems 3. Thermodynamics of metal–air systems 4. Combustion synthesis C. Diffusional kinetics D. Diffusion-controlled burning rate 1. Burning of metals in nearly pure oxygen 2. Burning of small particles – diffusion versus kinetic limits 3. The burning of boron particles 4. Carbon particle combustion (C. R. Shaddix) E. Practical carbonaceous fuels (C. R. Shaddix) 1. Devolatilization 2. Char combustion 3. Pulverized coal char oxidation 4. Gasification and oxy-combustion F. Soot oxidation (C. R. Shaddix) Problems

495 496 496 496 509 513 520 522 524 527 530 531 534 534 539 540 542 545 548

APPENDIXES

551

APPENDIX A. THERMOCHEMICAL DATA AND CONVERSION FACTORS

555

Table A1. Table A2. Table A3.

Conversion factors and physical constants Thermochemical data for selected chemical compounds Thermochemical data for species included in reaction list of Appendix C

556 557 646

APPENDIX B. ADIABATIC FLAME TEMPERATURES OF HYDROCARBONS

653

Table B1.

653

Adiabatic flame temperatures

xiv

Contents

APPENDIX C. SPECIFIC REACTION RATE CONSTANTS

659

Table C1. Table C2. Table C3. Table C4. Table C5. Table C6. Table C7. Table C8. Table C9. Table C10. Table C11.

659 661 662 663 665 668 673 677 683 684 685

H2/O2 mechanism CO/H2/O2 mechanism CH2O/CO/H2/O2 mechanism CH3OH/CH2O/CO/H2/O2 mechanism CH4/CH3OH/CH2O/CO/H2/O2 mechanism C2H6/CH4/CH3OH/CH2O/CO/H2/O2 mechanism Selected reactions of a C3H8 oxidation mechanism NxOy/CO/H2/O2 mechanism HCl/NxOy/CO/H2/O2 mechanism O3/NxOy/CO/H2/O2 mechanism SOx/NxOy/CO/H2/O2 mechanism

APPENDIX D. BOND DISSOCIATION ENERGIES OF HYDROCARBONS

693

Table D1. Bond dissociation energies of alkanes Table D2. Bond dissociation energies of alkenes, alkynes, and aromatics Table D3. Bond dissociation energies of C/H/O compounds Table D4. Bond dissociation energies of sulfur-containing compounds Table D5. Bond dissociation energies of nitrogen-containing compounds Table D6. Bond dissociation energies of halocarbons

694

APPENDIX E. FLAMMABILITY LIMITS IN AIR

703

Table E1.

Flammability limits of fuel gases and vapors in air at 25°C and 1 atm

APPENDIX F. LAMINAR FLAME SPEEDS Table F1.

Table F2.

Table F3.

Burning velocities of various fuels at 25°C air-fuel temperature (0.31 mol% H2O in air). Burning velocity S as a function of equivalence ratio φ in cm/s Burning velocities of various fuels at 100°C air-fuel temperature (0.31 mol% H2O in air). Burning velocity S as a function of equivalence ratio φ in cm/s Burning velocities of various fuels in air as a function of pressure for an equivalence ratio of 1 in cm/s

695 698 699 700 702

704

713

714

719 720

APPENDIX G. SPONTANEOUS IGNITION TEMPERATURE DATA

721

Table G1. Spontaneous ignition temperature data

722

Contents

xv

APPENDIX H. MINIMUM SPARK IGNITION ENERGIES AND QUENCHING DISTANCES

743

Table H1. Minimum spark ignition energy data for fuels in air at 1 atm pressure

744

APPENDIX I. PROGRAMS FOR COMBUSTION KINETICS

747

A. B. C. D. E. F. G. H. I. J. K. L. M.

747 747 748 748 750 752 753 754 754 756 756 756 756

Thermochemical parameters Kinetic parameters Transport parameters Reaction mechanisms Thermodynamic equilibrium Temporal kinetics (Static and flow reactors) Stirred reactors Shock tubes Premixed flames Diffusion flames Boundary layer flow Detonations Model analysis and mechanism reduction

Author Index

759

Subject Index

769

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Prologue

This 4th Edition of “Combustion” was initiated at the request of the publisher, but it was the willingness of Prof. Richard Yetter to assume the responsibility of co-author that generated the undertaking. Further, the challenge brought to mind the oversight of an acknowledgment that should have appeared in the earlier editions. After teaching the combustion course I developed at Princeton for 25 years, I received a telephone call in 1975 from Prof. Bill Reynolds, who at the time was Chairman of the Mechanical Engineering Department at Stanford. Because Stanford was considering developing combustion research, he invited me to present my Princeton combustion course during Stanford’s summer semester that year. He asked me to take in consideration that at the present time their graduate students had little background in combustion, and, further, he wished to have the opportunity to teleconference my presentation to Berkeley, Ames, and Sandia Livermore. It was an interesting challenge and I accepted the invitation as the Standard Oil of California Visiting Professor of Combustion. My early lectures seemed to receive a very favorable response from those participating in the course. Their only complaint was that there were no notes to help follow the material presented. Prof. Reynolds approached me with the request that a copy of lecture notes be given to all the attendees. He agreed it was not appropriate when he saw the handwritten copies from which I presented the lectures. He then proposed that I stop all other interactions with my Stanford colleagues during my stay and devote all my time to writing these notes in the proper grammatical and structural form. Further, to encourage my writing he would assign a secretary to me who would devote her time organizing and typing my newly written notes. Of course, the topic of a book became evident in the discussion. Indeed, eight of the nine chapters of the first edition were completed during this stay at Stanford and it took another 2 years to finish the last chapter, indexes, problems, etc., of this first edition. Thus I regret that I never acknowledged with many thanks to Prof. Reynolds while he was alive for being the spark that began the editions of “Combustion” that have already been published. “Combustion, 4th Edition” may appear very similar in format to the 3rd Edition. There are new sections and additions, and many brief insertions that are the core of important modifications. It is interesting that the content of these insertions emanated from an instance that occurred during my Stanford presentation. At one lecture, an attendee who obviously had some experience xvii

xviii

Prologue

in the combustion field claimed that I had left out certain terms that usually appear in one of the simple analytical developments I was discussing. Surprisingly, I subconscientiously immediately responded “You don’t swing at the baseball until you get to the baseball park!” The response, of course, drew laughter, but everyone appeared to understand the point I was trying to make. The reason of bringing up this incident is that it is important to develop the understanding of a phenomenon, rather than all its detailed aspects. I have always stressed to my students that there is a great difference between knowing something and understanding it. The relevant point is that in various sections there have been inserted many small, important modifications to give greater understanding to many elements of combustion that appear in the text. This type of material did not require extensive paragraphs in each chapter section. Most chapters in this edition contain, where appropriate, this type of important improvement. This new material and other major additions are self-evident in the listings in the Table of Contents. My particular thanks go to Prof. Yetter for joining me as co-author, for his analyzing and making small poignant modifications of the chapters that appeared in the earlier additions, for contributing new material not covered in these earlier additions and for further developing all the appendixes. Thanks also go to Dr. Chris Shaddix of Sandia Livermore who made a major contribution to Chapter 9 with respect to coal combustion considerations. Our gracious thanks go to Mary Newby of Penn State who saw to the final typing of the complete book and who offered a great deal of general help. We would never have made it without her. We also wish to thank our initial editor at Elsevier, Joel Stein, for convincing us to undertake this edition of “Combustion” and our final Editor, Matthew Hart, for seeing this endeavor through. The last acknowledgments go to all who are recognized in the Dedication. I initiated what I called Princeton’s Combustion Research Laboratory when I was first appointed to the faculty there and I am pleased that Prof. Fred Dryer now continues the philosophy of this laboratory. It is interesting to note that Profs. Dryer and Yetter and Dr. Shaddix were always partners of this laboratory from the time that they entered Princeton as graduate students. I thank them again for being excellent, thoughtful, and helpful colleagues through the years. Speaking for Prof. Yetter as well, our hope is that “Combustion, 4th Edition” will be a worthwhile contributing and useful endeavor. Irvin Glassman December 2007

Preface

When approached by the publisher Elsevier to consider writing a 4th Edition of Combustion, we considered the challenge was to produce a book that would extend the worthiness of the previous editions. Since the previous editions served as a basis of understanding of the combustion field, and as a text to be used in many class courses, we realized that, although the fundamentals do not change, there were three factors worthy of consideration: to add and extend all chapters so that the fundamentals could be clearly seen to provide the background for helping solve challenging combustion problems; to enlarge the Appendix section to provide even more convenient data tables and computational programs; and to enlarge the number of typical problem sets. More important is the attempt to have these three factors interact so that there is a deeper understanding of the fundamentals and applications of each chapter. Whether this concept has been successful is up to the judgment of the reader. Some partial examples of this approach in each chapter are given by what follows. Thus, Chapter 1, Chemical Thermodynamics and Flame Temperatures, is now shown to be important in understanding scramjets. Chapter 2, Chemical Kinetics, now explains how sensitivity analyses permit easier understanding in the analysis of complex reaction mechanisms that endeavor to explain environmental problems. There are additions and changes in Chapter 3, Explosive and General Oxidative Characteristics of Fuels, such as consideration of wet CO combustion analysis, the development procedure of reaction sensitivity analysis and the effect of supercritical conditions. Similarly the presentation in Chapter 4, Flame Phenomena in Premixed Combustible Gases, now considers flame propagation of stratified fuel–air mixtures and flame spread over liquid fuel spills. A point relevant to detonation engines has been inserted in Chapter 5. Chapter 6, Diffusion Flames, more carefully analyzes the differences between momentum and buoyant fuel jets. Ignition by pyrophoric materials, catalysts, and hypergolic fuels is now described in Chapter 7. The soot section in Chapter 8, Environmental Combustion Considerations, has been completely changed and also points out that most opposed jet diffusion flame experiments must be carefully analyzed since there is a difference between the temperature fields in opposed jet diffusion flames and simple fuel jets. Lastly, Chapter 9, Combustion of Nonvolatile Fuels, has a completely new approach to carbon combustion.

xix

xx

Preface

The use of the new material added to the Appendices should help students as the various new problem sets challenge them. Indeed, this approach has changed the character of the chapters that appeared in earlier editions regardless of apparent similarity in many cases. It is the hope of the authors that the objectives of this edition have been met. Irvin Glassman Richard A. Yetter

Chapter 1

Chemical Thermodynamics and Flame Temperatures A. INTRODUCTION The parameters essential for the evaluation of combustion systems are the equilibrium product temperature and composition. If all the heat evolved in the reaction is employed solely to raise the product temperature, this temperature is called the adiabatic flame temperature. Because of the importance of the temperature and gas composition in combustion considerations, it is appropriate to review those aspects of the field of chemical thermodynamics that deal with these subjects.

B. HEATS OF REACTION AND FORMATION All chemical reactions are accompanied by either an absorption or evolution of energy, which usually manifests itself as heat. It is possible to determine this amount of heat—and hence the temperature and product composition—from very basic principles. Spectroscopic data and statistical calculations permit one to determine the internal energy of a substance. The internal energy of a given substance is found to be dependent upon its temperature, pressure, and state and is independent of the means by which the state is attained. Likewise, the change in internal energy, ΔE, of a system that results from any physical change or chemical reaction depends only on the initial and final state of the system. Regardless of whether the energy is evolved as heat, energy, or work, the total change in internal energy will be the same. If a flow reaction proceeds with negligible changes in kinetic energy and potential energy and involves no form of work beyond that required for the flow, the heat added is equal to the increase of enthalpy of the system Q  ΔH where Q is the heat added and H is the enthalpy. For a nonflow reaction proceeding at constant pressure, the heat added is also equal to the gain in enthalpy Q  ΔH 1

2

Combustion

and if heat evolved, Q  ΔH Most thermochemical calculations are made for closed thermodynamic systems, and the stoichiometry is most conveniently represented in terms of the molar quantities as determined from statistical calculations. In dealing with compressible flow problems in which it is essential to work with open thermodynamic systems, it is best to employ mass quantities. Throughout this text uppercase symbols will be used for molar quantities and lowercase symbols for mass quantities. One of the most important thermodynamic facts to know about a given chemical reaction is the change in energy or heat content associated with the reaction at some specified temperature, where each of the reactants and products is in an appropriate standard state. This change is known either as the energy or as the heat of reaction at the specified temperature. The standard state means that for each state a reference state of the aggregate exists. For gases, the thermodynamic standard reference state is the ideal gaseous state at atmospheric pressure at each temperature. The ideal gaseous state is the case of isolated molecules, which give no interactions and obey the equation of state of a perfect gas. The standard reference state for pure liquids and solids at a given temperature is the real state of the substance at a pressure of 1 atm. As discussed in Chapter 9, understanding this definition of the standard reference state is very important when considering the case of high-temperature combustion in which the product composition contains a substantial mole fraction of a condensed phase, such as a metal oxide. The thermodynamic symbol that represents the property of the substance in the standard state at a given temperature is written, for example, as HT , ET , etc., where the “degree sign” superscript ° specifies the standard state, and the subscript T the specific temperature. Statistical calculations actually permit the determination of ET  E0, which is the energy content at a given temperature referred to the energy content at 0 K. For 1 mol in the ideal gaseous state, PV  RT

(1.1)

H   E   ( PV )  E   RT

(1.2)

H 0  E0

(1.3)

which at 0 K reduces to

Thus the heat content at any temperature referred to the heat or energy content at 0 K is known and ( H   H 0 )  ( E   E0 )  RT  ( E   E0 )  PV

(1.4)

3

Chemical Thermodynamics and Flame Temperatures

The value ( E   E0 ) is determined from spectroscopic information and is actually the energy in the internal (rotational, vibrational, and electronic) and external (translational) degrees of freedom of the molecule. Enthalpy ( H   H 0 ) has meaning only when there is a group of molecules, a mole for instance; it is thus the Ability of a group of molecules with internal energy to do PV work. In this sense, then, a single molecule can have internal energy, but not enthalpy. As stated, the use of the lowercase symbol will signify values on a mass basis. Since flame temperatures are calculated for a closed thermodynamic system and molar conservation is not required, working on a molar basis is most convenient. In flame propagation or reacting flows through nozzles, conservation of mass is a requirement for a convenient solution; thus when these systems are considered, the per unit mass basis of the thermochemical properties is used. From the definition of the heat of reaction, Qp will depend on the temperature T at which the reaction and product enthalpies are evaluated. The heat of reaction at one temperature T0 can be related to that at another temperature T1. Consider the reaction configuration shown in Fig. 1.1. According to the First Law of Thermodynamics, the heat changes that proceed from reactants at temperature T0 to products at temperature T1, by either path A or path B must be the same. Path A raises the reactants from temperature T0 to T1, and reacts at T1. Path B reacts at T0 and raises the products from T0 to T1. This energy equality, which relates the heats of reaction at the two different temperatures, is written as ⎧⎪ ⎡ ⎪ ⎨ ∑ n j ⎢ HT1  H 0  HT0  H 0 ⎪⎪ j ,react ⎣ ⎪⎩ ⎧ ⎡  ΔHT0  ⎪⎪⎨ ∑ ni ⎢ HT1  H 0 ⎪⎪⎩ i,prod ⎣

) (

(

(

⎫⎪

)⎤⎥⎦ ⎪⎬⎪⎪  ΔHH j

⎪⎭

)  ( H

T0

T1

⎤ ⎫  H 0 ⎥ ⎪⎪⎬ ⎦i ⎪ ⎪⎭

)

(1.5)

where n specifies the number of moles of the ith product or jth reactant. Any phase changes can be included in the heat content terms. Thus, by knowing the difference in energy content at the different temperatures for the products and

(1)

ΔHT1

(2)

T1

Path A

(1)

Path B

(2)

T0

ΔHT0 Reactants

Products

FIGURE 1.1 Heats of reactions at different base temperatures.

4

Combustion

reactants, it is possible to determine the heat of reaction at one temperature from the heat of reaction at another. If the heats of reaction at a given temperature are known for two separate reactions, the heat of reaction of a third reaction at the same temperature may be determined by simple algebraic addition. This statement is the Law of Heat Summation. For example, reactions (1.6) and (1.7) can be carried out conveniently in a calorimeter at constant pressure: Cgraphite  O2 (g) ⎯ ⎯⎯⎯ → CO2 (g), 298 K

Q p  393.52 kJ

(1.6)

CO(g)  21 O2 (g) ⎯ ⎯⎯⎯ → CO2 (g), 298 K

Q p  283.0 kJ

(1.7)

Q p  110.52 kJ

(1.8)

Subtracting these two reactions, one obtains Cgraphite  21 O2 (g) ⎯ ⎯⎯⎯ → CO(g), 298 K

Since some of the carbon would burn to CO2 and not solely to CO, it is difficult to determine calorimetrically the heat released by reaction (1.8). It is, of course, not necessary to have an extensive list of heats of reaction to determine the heat absorbed or evolved in every possible chemical reaction. A more convenient and logical procedure is to list the standard heats of formation of chemical substances. The standard heat of formation is the enthalpy of a substance in its standard state referred to its elements in their standard states at the same temperature. From this definition it is obvious that heats of formation of the elements in their standard states are zero. The value of the heat of formation of a given substance from its elements may be the result of the determination of the heat of one reaction. Thus, from the calorimetric reaction for burning carbon to CO2 [Eq. (1.6)], it is possible to write the heat of formation of carbon dioxide at 298 K as (ΔH f )298, CO2  393.52 kJ/mol The superscript to the heat of formation symbol ΔH f represents the standard state, and the subscript number represents the base or reference temperature. From the example for the Law of Heat Summation, it is apparent that the heat of formation of carbon monoxide from Eq. (1.8) is (ΔH f )298, CO  110.52 kJ/mol It is evident that, by judicious choice, the number of reactions that must be measured calorimetrically will be about the same as the number of substances whose heats of formation are to be determined.

5

Chemical Thermodynamics and Flame Temperatures

The logical consequence of the preceding discussion is that, given the heats of formation of the substances comprising any particular reaction, one can directly determine the heat of reaction or heat evolved at the reference temperature T0, most generally T298, as follows: ΔHT0 

∑

i prod

ni (ΔH f )T0 , i 

∑

j react

n j (ΔH f )T0 , j  Q p

(1.9)

Extensive tables of standard heats of formation are available, but they are not all at the same reference temperature. The most convenient are the compilations known as the JANAF [1] and NBS Tables [2], both of which use 298 K as the reference temperature. Table 1.1 lists some values of the heat of formation taken from the JANAF Thermochemical Tables. Actual JANAF tables are reproduced in Appendix A. These tables, which represent only a small selection from the JANAF volume, were chosen as those commonly used in combustion and to aid in solving the problem sets throughout this book. Note that, although the developments throughout this book take the reference state as 298 K, the JANAF tables also list ΔH f for all temperatures. When the products are measured at a temperature T2 different from the reference temperature T0 and the reactants enter the reaction system at a temperature T0 different from the reference temperature, the heat of reaction becomes ΔH 

{( H   H  )  ( H   H  )}  (ΔH  ) ⎤⎥⎥⎦ ⎡  ∑ n ⎢ {( H   H  )  ( H  H  )}  (ΔH  ) ⎢⎣

⎡ ni ⎢ ⎢⎣ i prod

∑

0

T2

j

j react

 Q p (evolved )

T0

0

f T0

0

T0

 T0

0

i

f T0

⎤ ⎥ ⎥⎦ j (1.10)

The reactants in most systems are considered to enter at the standard reference temperature 298 K. Consequently, the enthalpy terms in the braces for the reactants disappear. The JANAF tables tabulate, as a putative convenience,  ) instead of ( HT  H 0 ). This type of tabulation is unfortunate ( H T  H 298 since the reactants for systems using cryogenic fuels and oxidizers, such as those used in rockets, can enter the system at temperatures lower than the reference temperature. Indeed, the fuel and oxidizer individually could enter at different temperatures. Thus the summation in Eq. (1.10) is handled most conveniently by realizing that T0 may vary with the substance j. The values of heats of formation reported in Table 1.1 are ordered so that the largest positive values of the heats of formation per mole are the highest and those with negative heats of formation are the lowest. In fact, this table is similar to a potential energy chart. As species at the top react to form species at the bottom, heat is released, and an exothermic system exists. Even a species that has a negative heat of formation can react to form products of still lower negative heats of formation species, thereby releasing heat. Since some fuels that have

TABLE 1.1 Heats of Formation at 298 K

ΔH f ( kJ/mol)

Δhf ( kJ/g mol)

Chemical

Name

State

C

Carbon

Vapor

716.67

59.72

N

Nitrogen atom

Gas

472.68

33.76

O

Oxygen atom

Gas

249.17

15.57

C2H2

Acetylene

Gas

227.06

8.79

H

Hydrogen atom

Gas

218.00

218.00

O3

Ozone

Gas

142.67

2.97

NO

Nitric oxide

Gas

90.29

3.01

C6H6

Benzene

Gas

82.96

1.06

C6H6

Benzene

Liquid

49.06

0.63

C2H4

Ethene

Gas

52.38

1.87

N2H4

Hydrazine

Liquid

50.63

1.58

OH

Hydroxyl radical

Gas

38.99

2.29

O2

Oxygen

Gas

0

0

N2

Nitrogen

Gas

0

0

H2

Hydrogen

Gas

0

0

C

Carbon

Solid

0

0

NH3

Ammonia

Gas

45.90

2.70

C2H4O

Ethylene oxide

Gas

51.08

0.86

CH4

Methane

Gas

74.87

4.68

C2H6

Ethane

Gas

84.81

2.83

CO

Carbon monoxide

Gas

110.53

3.95

C4H10

Butane

Gas

124.90

2.15

CH3OH

Methanol

Gas

201.54

6.30

CH3OH

Methanol

Liquid

239.00

7.47

H2O

Water

Gas

241.83

13.44

C8H18

Octane

Liquid

250.31

0.46

H2O

Water

Liquid

285.10

15.84

SO2

Sulfur dioxide

Gas

296.84

4.64

C12H16

Dodecane

Liquid

347.77

2.17

CO2

Carbon dioxide

Gas

393.52

8.94

SO3

Sulfur trioxide

Gas

395.77

4.95

7

Chemical Thermodynamics and Flame Temperatures

negative heats of formation form many moles of product species having negative heats of formation, the heat release in such cases can be large. Equation (1.9) shows this result clearly. Indeed, the first summation in Eq. (1.9) is generally much greater than the second. Thus the characteristic of the reacting species or the fuel that significantly determines the heat release is its chemical composition and not necessarily its molar heat of formation. As explained in Section D2, the heats of formation listed on a per unit mass basis simplifies one’s ability to estimate relative heat release and temperature of one fuel to another without the detailed calculations reported later in this chapter and in Appendix I. The radicals listed in Table 1.1 that form their respective elements have their heat release equivalent to the radical’s heat of formation. It is then apparent that this heat release is also the bond energy of the element formed. Non-radicals such as acetylene, benzene, and hydrazine can decompose to their elements and/ or other species with negative heats of formation and release heat. Consequently, these fuels can be considered rocket monopropellants. Indeed, the same would hold for hydrogen peroxide; however, what is interesting is that ethylene oxide has a negative heat of formation, but is an actual rocket monopropellant because it essentially decomposes exothermically into carbon monoxide and methane [3]. Chemical reaction kinetics restricts benzene, which has a positive heat of formation from serving as a monopropellant because its energy release is not sufficient to continuously initiate decomposition in a volumetric reaction space such as a rocket combustion chamber. Insight into the fundamentals for understanding this point is covered in Chapter 2, Section B1. Indeed, for acetylene type and ethylene oxide monopropellants the decomposition process must be initiated with oxygen addition and spark ignition to then cause self-sustained decomposition. Hydrazine and hydrogen peroxide can be ignited and self-sustained with a catalyst in a relatively small volume combustion chamber. Hydrazine is used extensively for control systems, back pack rockets, and as a bipropellant fuel. It should be noted that in the Gordon and McBride equilibrium thermodynamic program [4] discussed in Appendix I, the actual results obtained might not be realistic because of kinetic reaction conditions that take place in the short stay times in rocket chambers. For example, in the case of hydrazine, ammonia is a product as well as hydrogen and nitrogen [5]. The overall heat release is greater than going strictly to its elements because ammonia is formed in the decomposition process and is frozen in its composition before exiting the chamber. Ammonia has a relatively large negative heat of formation. Referring back to Eq. (1.10), when all the heat evolved is used to raise the temperature of the product gases, ΔH and Qp become zero. The product temperature T2 in this case is called the adiabatic flame temperature and Eq. (1.10) becomes

{(

)}

⎡ ⎤ ni ⎢ HT  H 0  HT  H 0  (ΔH f )T0 ⎥ 2 0 ⎢⎣ ⎥⎦ i i prod ⎡ ⎤  ∑ n j ⎢ HT   H 0  HT  H 0  (ΔH f )T0 ⎥ 0 0 ⎥⎦ j ⎣⎢ j react

∑

) (

{(

) (

)}

(1.11)

8

Combustion

Again, note that T0 can be different for each reactant. Since the heats of formation throughout this text will always be considered as those evaluated at the reference temperature T0  298 K, the expression in braces becomes {( HT  H 0 )  ( HT  H 0 )}  ( HT  HT ), which is the value listed in the 0 0 JANAF tables (see Appendix A). If the products ni of this reaction are known, Eq. (1.11) can be solved for the flame temperature. For a reacting lean system whose product temperature is less than 1250 K, the products are the normal stable species CO2, H2O, N2, and O2, whose molar quantities can be determined from simple mass balances. However, most combustion systems reach temperatures appreciably greater than 1250 K, and dissociation of the stable species occurs. Since the dissociation reactions are quite endothermic, a small percentage of dissociation can lower the flame temperature substantially. The stable products from a C¶H¶O reaction system can dissociate by any of the following reactions: CO2  CO  21 O2 CO2  H 2  CO  H 2 O H 2 O  H 2  21 O2 H 2 O  H  OH H2 O 

1 2

H 2  OH

H 2  2H O2  2O, etc. Each of these dissociation reactions also specifies a definite equilibrium concentration of each product at a given temperature; consequently, the reactions are written as equilibrium reactions. In the calculation of the heat of reaction of low-temperature combustion experiments the products could be specified from the chemical stoichiometry; but with dissociation, the specification of the product concentrations becomes much more complex and the ni’s in the flame temperature equation [Eq. (1.11)] are as unknown as the flame temperature itself. In order to solve the equation for the ni’s and T2, it is apparent that one needs more than mass balance equations. The necessary equations are found in the equilibrium relationships that exist among the product composition in the equilibrium system.

C. FREE ENERGY AND THE EQUILIBRIUM CONSTANTS The condition for equilibrium is determined from the combined form of the first and second laws of thermodynamics; that is, dE  TdS  PdV

(1.12)

9

Chemical Thermodynamics and Flame Temperatures

where S is the entropy. This condition applies to any change affecting a system of constant mass in the absence of gravitational, electrical, and surface forces. However, the energy content of the system can be changed by introducing more mass. Consider the contribution to the energy of the system on adding one molecule i to be μi. The introduction of a small number dni of the same type contributes a gain in energy of the system of μi dni. All the possible reversible increases in the energy of the system due to each type of molecule i can be summed to give dE  TdS  PdV  ∑ μi dni

(1.13)

i

It is apparent from the definition of enthalpy H and the introduction of the concept of the Gibbs free energy G G  H  TS

(1.14)

that dH  TdS  VdP  ∑ μi dni

(1.15)

dG  SdT  VdP  ∑ μi dni

(1.16)

i

and

i

Recall that P and T are intensive properties that are independent of the size of mass of the system, whereas E, H, G, and S (as well as V and n) are extensive properties that increase in proportion to mass or size. By writing the general relation for the total derivative of G with respect to the variables in Eq. (1.16), one obtains ⎛ ∂G ⎞⎟ ⎛ ∂G ⎞⎟ ⎛ ∂G ⎞⎟ ⎟⎟ dG  ⎜⎜ dT  ⎜⎜ ⎟⎟ ⎟⎟ dP  ∑ ⎜⎜⎜ ⎜⎝ ∂T ⎠ ⎜⎝ ∂P ⎠ ⎝ ∂ni ⎟⎠P ,T, n i ⎜ P, n T, n i

i

dni

(1.17)

j ( j i )

Thus, ⎛ ∂G ⎞⎟ ⎟ μi  ⎜⎜⎜ ⎜⎝ ∂ni ⎟⎟⎠ T,P,n

(1.18) j

or, more generally, from dealing with the equations for E and H ⎛ ∂E ⎞⎟ ⎛ ∂G ⎞⎟ ⎛ ∂H ⎞⎟ ⎟⎟ ⎟⎟ ⎟ μi  ⎜⎜⎜  ⎜⎜⎜  ⎜⎜⎜ ⎜⎝ ∂ni ⎟⎠ ⎜⎝ ∂ni ⎟⎠ ⎜⎝ ∂ni ⎟⎟⎠ T, P, n S ,V,n S ,P, n j

j

(1.19) j

10

Combustion

where μi is called the chemical potential or the partial molar free energy. The condition of equilibrium is that the entropy of the system have a maximum value for all possible configurations that are consistent with constant energy and volume. If the entropy of any system at constant volume and energy is at its maximum value, the system is at equilibrium; therefore, in any change from its equilibrium state dS is zero. It follows then from Eq. (1.13) that the condition for equilibrium is

∑ μi dni  0

(1.20)

The concept of the chemical potential is introduced here because this property plays an important role in reacting systems. In this context, one may consider that a reaction moves in the direction of decreasing chemical potential, reaching equilibrium only when the potential of the reactants equals that of the products [3]. Thus, from Eq. (1.16) the criterion for equilibrium for combustion products of a chemical system at constant T and P is (dG )T, P  0

(1.21)

and it becomes possible to determine the relationship between the Gibbs free energy and the equilibrium partial pressures of a combustion product mixture. One deals with perfect gases so that there are no forces of interactions between the molecules except at the instant of reaction; thus, each gas acts as if it were in a container alone. Let G, the total free energy of a product mixture, be represented by G  ∑ ni Gi ,

i  A, B, … , R, S...

(1.22)

for an equilibrium reaction among arbitrary products: aA  bB  …  rR  sS  …

(1.23)

Note that A, B, …, R, S, … represent substances in the products only and a, b, …, r, s, … are the stoichiometric coefficients that govern the proportions by which different substances appear in the arbitrary equilibrium system chosen. The ni’s represent the instantaneous number of each compound. Under the ideal gas assumption the free energies are additive, as shown above. This assumption permits one to neglect the free energy of mixing. Thus, as stated earlier, G( P, T )  H (T )  TS ( P, T )

(1.24)

Since the standard state pressure for a gas is P0  1 atm, one may write G ( P0 , T )  H (T )  TS ( P0 , T )

(1.25)

11

Chemical Thermodynamics and Flame Temperatures

Subtracting the last two equations, one obtains G  G   ( H  H )  T (S  S )

(1.26)

Since H is not a function of pressure, H  H° must be zero, and then G  G   T (S  S )

(1.27)

Equation (1.27) relates the difference in free energy for a gas at any pressure and temperature to the standard state condition at constant temperature. Here dH  0, and from Eq. (1.15) the relationship of the entropy to the pressure is found to be S  S   R ln( p/p0 )

(1.28)

G(T , P )  G   RT ln( p/p0 )

(1.29)

Hence, one finds that

An expression can now be written for the total free energy of a gas mixture. In this case P is the partial pressure Pi of a particular gaseous component and obviously has the following relationship to the total pressure P: ⎛ n ⎞ pi  ⎜⎜ i ⎟⎟⎟ P ⎜⎜ ∑ n ⎟ i ⎟ ⎜⎝ ⎟⎠ i

(1.30)

where (ni /∑ i ni ) is the mole fraction of gaseous species i in the mixture. Equation (1.29) thus becomes

{

G(T , P )  ∑ ni Gi  RT ln(pi /p0 ) i

}

(1.31)

As determined earlier [Eq. (1.21)], the criterion for equilibrium is (dG)T,P  0. Taking the derivative of G in Eq. (1.31), one obtains

∑ Gi dni  RT ∑ (dni ) ln( pi /p0 )  RT ∑ ni (dpi /pi )  0 i

i

(1.32)

i

Evaluating the last term of the left-hand side of Eq. (1.32), one has

∑ ni i

⎛ ∑ ni ⎞⎟ dpi ⎜  ∑ ⎜⎜ i ⎟⎟⎟ dpi  ⎜ p ⎟⎠ pi i ⎝

∑ i ni p

∑ dpi  0 i

(1.33)

12

Combustion

since the total pressure is constant, and thus first term in Eq. (1.32):

∑ i dpi

 0. Now consider the

∑ Gi dni  (dnA )GA  (dnB )GB (dnR )GR  (dnS )Gs 

(1.34)

i

By the definition of the stoichiometric coefficients, dni ~ ai ,

dni  kai

(1.35)

where k is a proportionality constant. Hence

∑ Gi dni  k{aGA  bGB rGR  sGS }

(1.36)

i

Similarly, the proportionality constant k will appear as a multiplier in the second term of Eq. (1.32). Since Eq. (1.32) must equal zero, the third term already has been shown equal to zero, and k cannot be zero, one obtains ⎪⎧ ( p /p )r ( p /p )s ⎪⎫ (aGA  bGB rGR  sGs )  RT ln ⎪⎨ R 0 a S 0 b ⎪⎬ (1.37) ⎪⎪ ( pA /p0 ) ( pB /p0 ) ⎪⎪ ⎭ ⎩ One then defines ΔG   aGA  bGB rGR  sGS 

(1.38)

where ΔG° is called the standard state free energy change and p0  1 atm. This name is reasonable since ΔG° is the change of free energy for reaction (1.23) if it takes place at standard conditions and goes to completion to the right. Since the standard state pressure p0 is 1 atm, the condition for equilibrium becomes ΔG   RT ln( prR p sS /p aA pbB )

(1.39)

where the partial pressures are measured in atmospheres. One then defines the equilibrium constant at constant pressure from Eq. (1.39) as K p  prR p sS /p aA pbB Then ΔG   RT ln K p ,

K p  exp(ΔG  /RT )

(1.40)

Chemical Thermodynamics and Flame Temperatures

13

where Kp is not a function of the total pressure, but rather a function of temperature alone. It is a little surprising that the free energy change at the standard state pressure (1 atm) determines the equilibrium condition at all other pressures. Equations (1.39) and (1.40) can be modified to account for nonideality in the product state; however, because of the high temperatures reached in combustion systems, ideality can be assumed even under rocket chamber pressures. The energy and mass conservation equations used in the determination of the flame temperature are more conveniently written in terms of moles; thus, it is best to write the partial pressure in Kp in terms of moles and the total pressure P. This conversion is accomplished through the relationship between partial pressure p and total pressure P, as given by Eq. (1.30). Substituting this expression for pi [Eq. (1.30)] in the definition of the equilibrium constant [Eq. (1.40)], one obtains K p  (nRr nSs /nAa nBb )( P/ ∑ ni )r sab

(1.41)

which is sometimes written as K p  K N ( P/ ∑ ni )r sab

(1.42)

K N  nRr nSs /nAa nBb

(1.43)

rsab  0

(1.44)

where

When

the equilibrium reaction is said to be pressure-insensitive. Again, however, it is worth repeating that Kp is not a function of pressure; however, Eq. (1.42) shows that KN can be a function of pressure. The equilibrium constant based on concentration (in moles per cubic centimeter) is sometimes used, particularly in chemical kinetic analyses (to be discussed in the next chapter). This constant is found by recalling the perfect gas law, which states that PV  ∑ ni RT

(1.45)

( P/ ∑ ni )  ( RT/V )

(1.46)

or

where V is the volume. Substituting for ( P/ ∑ ni ) in Eq. (1.42) gives ⎛ RT ⎞⎟r sab K p  ⎡⎢⎣ (nRr nSs ) /(nAa nBb ) ⎤⎦⎥ ⎜⎜ ⎟ ⎜⎝ V ⎟⎠

(1.47)

14

Combustion

or Kp 

(nR /V )r (nS /V )s ( RT )r + sab (nA /V)a (nB /V)b

(1.48)

Equation (1.48) can be written as K p  (CRr CSs /CAa CBb )( RT )r sab

(1.49)

where C  n /V is a molar concentration. From Eq. (1.49) it is seen that the definition of the equilibrium constant for concentration is KC  CRr CSs /CAa CBb

(1.50)

KC is a function of pressure, unless r  s  ab  0. Given a temperature and pressure, all the equilibrium constants (Kp, KN, and KC) can be determined thermodynamically from ΔG° for the equilibrium reaction chosen. How the equilibrium constant varies with temperature can be of importance. Consider first the simple derivative d (G/T ) T (dG/dT ) − G  dT T2

(1.51)

Recall that the Gibbs free energy may be written as G  E  PV  TS

(1.52)

dG dE dV dS  P S T dT dT dT dT

(1.53)

or, at constant pressure,

At equilibrium from Eq. (1.12) for the constant pressure condition T

dS dE dV  P dT dT dT

(1.54)

Combining Eqs. (1.53) and (1.54) gives, dG  S dT

(1.55)

d (G/T ) TS  G H   2 2 dT T T

(1.56)

Hence Eq. (1.51) becomes

15

Chemical Thermodynamics and Flame Temperatures

This expression is valid for any substance under constant pressure conditions. Applying it to a reaction system with each substance in its standard state, one obtains d (ΔG T )  (ΔH T 2 )dT

(1.57)

where ΔH° is the standard state heat of reaction for any arbitrary reaction aA  bB  → rR  sS  at temperature T (and, of course, a pressure of 1 atm). Substituting the expression for ΔG° given by Eq. (1.40) into Eq. (1.57), one obtains d ln K p /dT  ΔH RT 2

(1.58)

If it is assumed that ΔH° is a slowly varying function of T, one obtains ⎛K ⎜ p ln ⎜⎜ 2 ⎜⎜⎝ K p 1

⎞⎟ ⎛ ⎞ ⎟⎟   ΔH  ⎜⎜ 1  1 ⎟⎟ ⎟ ⎟⎟ R ⎜⎜⎝ T2 T1 ⎟⎠ ⎠

(1.59)

Thus for small changes in T

(Kp ) (Kp ) 2

1

T2 T1

when

In the same context as the heat of formation, the JANAF tables have tabulated most conveniently the equilibrium constants of formation for practically every substance of concern in combustion systems. The equilibrium constant of formation (Kp,f) is based on the equilibrium equation of formation of a species from its elements in their normal states. Thus by algebraic manipulation it is possible to determine the equilibrium constant of any reaction. In flame temperature calculations, by dealing only with equilibrium constants of formation, there is no chance of choosing a redundant set of equilibrium reactions. Of course, the equilibrium constant of formation for elements in their normal state is one. Consider the following three equilibrium reactions of formation:

H 2  21 O2  H 2 O,

K p, f( H2 O ) 

pH2 O

( pH )( pO )

1/2

2

1 2

H 2  H,

K p,f(H) 

2

pH

( pH )

1/2

2

1 2

O2  21 H 2  OH,

K p,f(OH) 

pOH 1/2

( pO ) ( pH ) 2

1/2

2

16

Combustion

The equilibrium reaction is always written for the formation of one mole of the substances other than the elements. Now if one desires to calculate the equilibrium constant for reactions such as H 2 O  H  OH

H2 O 

and

1 2

H 2  OH

one finds the respective Kp’s from K p, f(H) K p, f(OH) p p K p  H OH  , pH2 O K p, f ( H 2 O )

( pH )

1/2

Kp 

2

pOH

pH2 O



K p, f(OH) K p, f(H2 O)

Because of this type of result and the thermodynamic expression ΔG   RT ln K p the JANAF tables list log Kp,f. Note the base 10 logarithm. For those compounds that contain carbon and a combustion system in which solid carbon is found, the thermodynamic handling of the Kp is somewhat more difficult. The equilibrium reaction of formation for CO2 would be Cgraphite  O2  CO2 ,

Kp 

pCO2 pO2 pC

However, since the standard state of carbon is the condensed state, carbon graphite, the only partial pressure it exerts is its vapor pressure (pvp), a known thermodynamic property that is also a function of temperature. Thus, the preceding formation expression is written as K p (T ) pvp,C (T ) 

pCO2 pO2

 K p

The Kp,f ’s for substances containing carbon tabulated by JANAF are in reality K p , and the condensed phase is simply ignored in evaluating the equilibrium expression. The number of moles of carbon (or any other condensed phase) is not included in the ∑ n j since this summation is for the gas phase components contributing to the total pressure.

D. FLAME TEMPERATURE CALCULATIONS 1. Analysis If one examines the equation for the flame temperature [Eq. (1.11)], one can make an interesting observation. Given the values in Table 1.1 and the realization

Chemical Thermodynamics and Flame Temperatures

17

that many moles of product form for each mole of the reactant fuel, one can see that the sum of the molar heats of the products will be substantially greater than the sum of the molar heats of the reactants; that is,

∑

i prod

ni (ΔH f )i

∑

n j (ΔH f ) j

j react

Consequently, it would appear that the flame temperature is determined not by the specific reactants, but only by the atomic ratios and the specific atoms that are introduced. It is the atoms that determine what products will form. Only ozone and acetylene have positive molar heats of formation high enough to cause a noticeable variation (rise) in flame temperature. Ammonia has a negative heat of formation low enough to lower the final flame temperature. One can normalize for the effects of total moles of products formed by considering the heats of formation per gram (Δhf ); these values are given for some fuels and oxidizers in Table 1.1. The variation of (Δhf ) among most hydrocarbon fuels is very small. This fact will be used later in correlating the flame temperatures of hydrocarbons in air. One can draw the further conclusion that the product concentrations are also functions only of temperature, pressure, and the C/H/O ratio and not the original source of atoms. Thus, for any C¶H¶O system, the products will be the same; i.e., they will be CO2, H2O, and their dissociated products. The dissociation reactions listed earlier give some of the possible “new” products. A more complete list would be CO2 , H 2 O, CO, H 2 , O2 , OH, H, O, O3 , C, CH 4 For a C, H, O, N system, the following could be added: N 2 , N, NO, NH3 , NO , e Nitric oxide has a very low ionization potential and could ionize at flame temperatures. For a normal composite solid propellant containing C¶H¶O¶N¶Cl¶Al, many more products would have to be considered. In fact if one lists all the possible number of products for this system, the solution to the problem becomes more difficult, requiring the use of advanced computers and codes for exact results. However, knowledge of thermodynamic equilibrium constants and kinetics allows one to eliminate many possible product species. Although the computer codes listed in Appendix I essentially make it unnecessary to eliminate any product species, the following discussion gives one the opportunity to estimate which products can be important without running any computer code.

18

Combustion

Consider a C¶H¶O¶N system. For an overoxidized case, an excess of oxygen converts all the carbon and hydrogen present to CO2 and H2O by the following reactions: CO2  CO  21 O2 ,

Q p  283.2 kJ

H 2 O  H 2  21 O2 ,

Q p  242.2 kJ

H 2 O  H  OH,

Q p  284.5 kJ

where the Qp’s are calculated at 298 K. This heuristic postulate is based upon the fact that at these temperatures and pressures at least 1% dissociation takes place. The pressure enters into the calculations through Le Chatelier’s principle that the equilibrium concentrations will shift with the pressure. The equilibrium constant, although independent of pressure, can be expressed in a form that contains the pressure. A variation in pressure shows that the molar quantities change. Since the reactions noted above are quite endothermic, even small concentration changes must be considered. If one initially assumes that certain products of dissociation are absent and calculates a temperature that would indicate 1% dissociation of the species, then one must reevaluate the flame temperature by including in the product mixture the products of dissociation; that is, one must indicate the presence of CO, H2, and OH as products. Concern about emissions from power plant sources has raised the level of interest in certain products whose concentrations are much less than 1%, even though such concentrations do not affect the temperature even in a minute way. The major pollutant of concern in this regard is nitric oxide (NO). To make an estimate of the amount of NO found in a system at equilibrium, one would use the equilibrium reaction of formation of NO 1 2

N 2  21 O2  NO

As a rule of thumb, any temperature above 1700 K gives sufficient NO to be of concern. The NO formation reaction is pressure-insensitive, so there is no need to specify the pressure. If in the overoxidized case T2 2400 K at P  1 atm and T2 2800 K at P  20 atm, the dissociation of O2 and H2 becomes important; namely, H 2  2H, O2  2O,

Q p  436.6 kJ Q p  499.0 kJ

Although these dissociation reactions are written to show the dissociation of one mole of the molecule, recall that the Kp,f ’s are written to show the formation of one mole of the radical. These dissociation reactions are highly endothermic, and even very small percentages can affect the final temperature. The new products are H and O atoms. Actually, the presence of O atoms could be

Chemical Thermodynamics and Flame Temperatures

19

attributed to the dissociation of water at this higher temperature according to the equilibrium step H 2 O  H 2  O,

Q p  498.3 kJ

Since the heat absorption is about the same in each case, Le Chatelier’s principle indicates a lack of preference in the reactions leading to O. Thus in an overoxidized flame, water dissociation introduces the species H2, O2, OH, H, and O. At even higher temperatures, the nitrogen begins to take part in the reactions and to affect the system thermodynamically. At T 3000 K, NO forms mostly from the reaction 1 2

N 2  21 O2  NO,

Q p  90.5 kJ

rather than 1 2

N 2  H 2 O  NO  H 2 ,

Q p  332.7 kJ

If T2 3500 K at P  1 atm or T 3600 K at 20 atm, N2 starts to dissociate by another highly endothermic reaction: N 2  2 N,

Q p  946.9 kJ

Thus the complexity in solving for the flame temperature depends on the number of product species chosen. For a system whose approximate temperature range is known, the complexity of the system can be reduced by the approach discussed earlier. Computer programs and machines are now available that can handle the most complex systems, but sometimes a little thought allows one to reduce the complexity of the problem and hence the machine time. Equation (1.11) is now examined closely. If the ni’s (products) total a number μ, one needs (μ  1) equations to solve for the μ ni’s and T2. The energy equation is available as one equation. Furthermore, one has a mass balance equation for each atom in the system. If there are α atoms, then (μ  α) additional equations are required to solve the problem. These (μ  α) equations come from the equilibrium equations, which are basically nonlinear. For the C¶H¶O¶N system one must simultaneously solve five linear equations and (μ  4) nonlinear equations in which one of the unknowns, T2, is not even present explicitly. Rather, it is present in terms of the enthalpies of the products. This set of equations is a difficult one to solve and can be done only with modern computational codes.

20

Combustion

Consider the reaction between octane and nitric acid taking place at a pressure P as an example. The stoichiometric equation is written as nC8H18 C8 H18  nHNO3 HNO3 → nCO2 CO2  nH2 O H 2 O  nH2 H 2  nCO CO  nO2 O2  nN2 N 2  nOH OH  nNO NO  nO O  nC Csolid  nH H Since the mixture ratio is not specified explicitly for this general expression, no effort is made to eliminate products and μ  11. Thus the new mass balance equations (α  4) are NH NO NN NC

 2 nH2  2 nO2  2 nN2  nCO2

 2 nH2 O  nH  nH2 O  2 nCO2  nCO  nOH  nO  nNO  nNO  nCO  nC

where NH NO NC NN

 18nC8H18  nHNO3  3nHNO3  8nC8H18  nHNO3

The seven (μ  α  4  7) equilibrium equations needed would be nCO2

(i)

C  O2  CO2 ,

K p, f 

(ii)

H 2  21 O2  H 2 O,

⎛ nH O ⎞⎟ ⎛ P ⎞1/2 2 ⎟ ⎟⎟ ⎜⎜ K p, f  ⎜⎜⎜ ⎟⎟ ⎜⎜⎝ ∑ n ⎟⎠⎟⎟ ⎜⎜⎝ nH n1/2 i 2 O2 ⎠

(iii)

C  21 O2  CO,

⎛ n ⎞⎟ ⎛ P ⎞⎟1/ 2 ⎟⎜ K p, f  ⎜⎜ CO ⎜⎜ n1/2 ⎟⎟⎟ ⎜⎜⎜⎝ ∑ n ⎟⎟⎟⎠ i ⎝ O2 ⎠

(iv)

1 2

H 2  21 O2  OH,

⎞ ⎛ n K p, f  ⎜⎜ 1/2OH1/2 ⎟⎟⎟ ⎜⎜ n n ⎟⎟ ⎝ H 2 O2 ⎠

(v)

1 2

O2  21 N 2  NO,

K p, f 

nO2

nNO n1O/ 2 n1N/ 2 2

2

Chemical Thermodynamics and Flame Temperatures

(vi)

1 2

O2  O,

K p,f 

nO n1O/ 2 2

(vii)

1 2

H 2  H,

21

⎛ P ⎞⎟1/ 2 ⎜⎜ ⎟ ⎜⎜⎝ ∑ ni ⎟⎟⎠

⎛ n ⎞ ⎛ P ⎞⎟1/ 2 K p, f  ⎜⎜ 1H/ 2 ⎟⎟⎟ ⎜⎜ ⎜⎜ n ⎟⎟ ⎜⎜⎝ ∑ n ⎟⎟⎟⎠ i ⎝ H2 ⎠

In these equations ∑ ni includes only the gaseous products; that is, it does not include nC. One determines nC from the equation for NC. The reaction between the reactants and products is considered irreversible, so that only the products exist in the system being analyzed. Thus, if the reactants were H2 and O2, H2 and O2 would appear on the product side as well. In dealing with the equilibrium reactions, one ignores the molar quantities of the reactants H2 and O2. They are given or known quantities. The amounts of H2 and O2 in the product mixture would be unknowns. This point should be considered carefully, even though it is obvious. It is one of the major sources of error in first attempts to solve flame temperature problems by hand. There are various mathematical approaches for solving these equations by numerical methods [4, 6, 7]. The most commonly used program is that of Gordon and McBride [4] described in Appendix I. As mentioned earlier, to solve explicitly for the temperature T2 and the product composition, one must consider α mass balance equations, (μ  α) nonlinear equilibrium equations, and an energy equation in which one of the unknowns T2 is not even explicitly present. Since numerical procedures are used to solve the problem on computers, the thermodynamic functions are represented in terms of power series with respect to temperature. In the general iterative approach, one first determines the equilibrium state for the product composition at an initially assumed value of the temperature and pressure, and then one checks to see whether the energy equation is satisfied. Chemical equilibrium is usually described by either of two equivalent formulations—equilibrium constants or minimization of free energy. For such simple problems as determining the decomposition temperature of a monopropellant having few exhaust products or examining the variation of a specific species with temperature or pressure, it is most convenient to deal with equilibrium constants. For complex problems the problem reduces to the same number of interactive equations whether one uses equilibrium constants or minimization of free energy. However, when one uses equilibrium constants, one encounters more computational bookkeeping, more numerical difficulties with the use of components, more difficulty in testing for the presence of some condensed species, and more difficulty in extending the generalized methods to conditions that require nonideal equations of state [4, 6, 8]. The condition for equilibrium may be described by any of several thermodynamic functions, such as the minimization of the Gibbs or Helmholtz

22

Combustion

free energy or the maximization of entropy. If one wishes to use temperature and pressure to characterize a thermodynamic state, one finds that the Gibbs free energy is most easily minimized, inasmuch as temperature and pressure are its natural variables. Similarly, the Helmholtz free energy is most easily minimized if the thermodynamic state is characterized by temperature and volume (density) [4]. As stated, the most commonly used procedure for temperature and composition calculations is the versatile computer program of Gordon and McBride [4], who use the minimization of the Gibbs free energy technique and a descent Newton–Raphson method to solve the equations iteratively. A similar method for solving the equations when equilibrium constants are used is shown in Ref. [7].

2. Practical Considerations The flame temperature calculation is essentially the solution to a chemical equilibrium problem. Reynolds [8] has developed a more versatile approach to the solution. This method uses theory to relate mole fractions of each species to quantities called element potentials: There is one element potential for each independent atom in the system, and these element potentials, plus the number of moles in each phase, are the only variables that must be adjusted for the solution. In large problems there is a much smaller number than the number of species, and hence far fewer variables need to be adjusted. [8]

The program, called Stanjan [8] (see Appendix I), is readily handled even on the most modest computers. Like the Gordon–McBride program, both approaches use the JANAF thermochemical database [1]. The suite of CHEMKIN programs (see Appendix H) also provides an equilibrium code based on Stanjan [8]. In combustion calculations, one primarily wants to know the variation of the temperature with the ratio of oxidizer to fuel. Therefore, in solving flame temperature problems, it is normal to take the number of moles of fuel as 1 and the number of moles of oxidizer as that given by the oxidizer/fuel ratio. In this manner the reactant coefficients are 1 and a number normally larger than 1. Plots of flame temperature versus oxidizer/fuel ratio peak about the stoichiometric mixture ratio, generally (as will be discussed later) somewhat on the fuel-rich side of stoichiometric. If the system is overoxidized, the excess oxygen must be heated to the product temperature; thus, the product temperature drops from the stoichiometric value. If too little oxidizer is present—that is, the system is underoxidized—there is not enough oxygen to burn all the carbon and hydrogen to their most oxidized state, so the energy released is less and the temperature drops as well. More generally, the flame temperature is plotted as a function of the equivalence ratio (Fig. 1.2), where the equivalence ratio is defined as the fuel/oxidizer ratio divided by the stoichiometric fuel/oxidizer ratio. The equivalence ratio is given the symbol φ. For fuel-rich systems, there is more than the stoichiometric amount of fuel, and φ 1. For overoxidized,

23

Flame temperature

Chemical Thermodynamics and Flame Temperatures

φ<1 φ>1 1.0 fuel-lean fuel-rich φ  (F/A)/(F/A)STOICH FIGURE 1.2 Variation of flame temperature with equivalence ratio φ.

or fuel-lean systems, φ 1. Obviously, at the stoichiometric amount, φ  1. Since most combustion systems use air as the oxidizer, it is desirable to be able to conveniently determine the flame temperature of any fuel with air at any equivalence ratio. This objective is possible given the background developed in this chapter. As discussed earlier, Table 1.1 is similar to a potential energy diagram in that movement from the top of the table to products at the bottom indicates energy release. Moreover, as the size of most hydrocarbon fuel molecules increases, so does its negative heat of formation. Thus, it is possible to have fuels whose negative heats of formation approach that of carbon dioxide. It would appear, then, that heat release would be minimal. Heats of formation of hydrocarbons range from 227.1 kJ/mol for acetylene to 456.3 kJ/mol for n-ercosane (C20H42). However, the greater the number of carbon atoms in a hydrocarbon fuel, the greater the number of moles of CO2, H2O, and, of course, their formed dissociation products. Thus, even though a fuel may have a large negative heat of formation, it may form many moles of combustion products without necessarily having a low flame temperature. Then, in order to estimate the contribution of the heat of formation of the fuel to the flame temperature, it is more appropriate to examine the heat of formation on a unit mass basis rather than a molar basis. With this consideration, one finds that practically every hydrocarbon fuel has a heat of formation between 1.5 and 1.0 kcal/g. In fact, most fall in the range 2.1 to 2.1 kcal/g. Acetylene and methyl acetylene are the only exceptions, with values of 2.90 and 4.65 kcal/g, respectively. In considering the flame temperatures of fuels in air, it is readily apparent that the major effect on flame temperature is the equivalence ratio. Of almost equal importance is the H/C ratio, which determines the ratio of water vapor, CO2, and their formed dissociation products. Since the heats of formation per unit mass of olefins do not vary much and the H/C ratio is the same for all, it is not surprising that flame temperature varies little among the monoolefins.

24

Combustion

When discussing fuel-air mixture temperatures, one must always recall the presence of the large number of moles of nitrogen. With these conceptual ideas it is possible to develop simple graphs that give the adiabatic flame temperature of any hydrocarbon fuel in air at any equivalence ratio [9]. Such graphs are shown in Figs 1.3, 1.4, and 1.5. These graphs depict the flame temperatures for a range of hypothetical hydrocarbons that have heats of formation ranging from 1.5 to 1.0 kcal/g (i.e., from 6.3 to 4.2 kJ/g). The hydrocarbons chosen have the formulas CH4, CH3, CH2.5, CH2, CH1.5, and CH1; that is, they have H/C ratios of 4, 3, 2.5, 2.0, 1.5, and 1.0. These values include every conceivable hydrocarbon, except the acetylenes. The values listed, which were calculated from the standard Gordon–McBride computer program, were determined for all species entering at 298 K for a pressure of 1 atm. As a matter of interest, also plotted in the figures are the values of CH0, or a H/C ratio of 0. Since the only possible species with this H/C ratio is carbon, the only meaningful points from a physical point of view are those for a heat of formation of 0. The results in the figures plot the flame temperature as a function of the chemical enthalpy content of the reacting system in kilocalories per gram of reactant fuel. Conversion to kilojoules per gram can be made readily. In the figures there are lines of constant H/C ratio grouped according to the equivalence ratio φ. For most systems the enthalpy used as the abscissa will be the heat of formation of the fuel in kilocalories per gram, but there is actually greater versatility in using this enthalpy. For example, in a cooled flat flame burner, the measured heat extracted by the water can be converted on a unit fuel flow basis to a reduction in the heat of formation of the fuel. This lower enthalpy value is then used in the graphs to determine the adiabatic flame temperature. The same kind of adjustment can be made to determine the flame temperature when either the fuel or the air or both enter the system at a temperature different from 298 K. If a temperature is desired at an equivalence ratio other than that listed, it is best obtained from a plot of T versus φ for the given values. The errors in extrapolating in this manner or from the graph are trivial, less than 1%. The reason for separate Figs 1.4 and 1.5 is that the values for φ  1.0 and φ  1.1 overlap to a great extent. For Fig. 1.5, φ  1.1 was chosen because the flame temperature for many fuels peaks not at the stoichiometric value, but between φ  1.0 and 1.1 owing to lower mean specific heats of the richer products. The maximum temperature for acetylene-air peaks, for example, at a value of φ  1.3 (see Table 1.2). The flame temperature values reported in Fig. 1.3 show some interesting trends. The H/C ratio has a greater effect in rich systems. One can attribute this trend to the fact that there is less nitrogen in the rich cases as well as to a greater effect of the mean specific heat of the combustion products. For richer systems the mean specific heat of the product composition is lower owing to the preponderance of the diatomic molecules CO and H2 in comparison to the triatomic molecules CO2 and H2O. The diatomic molecules have lower

φ 1.1 1.0

2500 2400

1.5

2300 2200

0.75 2100 2.0 2000

Temperature (K)

1900 1800 1700 0.5 1600 1500 1400 1300 1200 1100 1000 1.5

1.0

0.0 0.5 Enthalpy (kcal/g)

0.5

1.0

FIGURE 1.3 Flame temperatures (in kelvins) of hydrocarbons and air as a function of the total enthalpy content of reactions (in kilocalories per gram) for various equivalence and H/C ratios at 1 atm pressure. Reference sensible enthalpy related to 298 K. The H/C ratios are in the following order: —--— —— …… —— —-— ——— … …

H/C  4 H/C  3 H/C  2.5 H/C  2.0 H/C  1.5 H/C  1.0 H/C  0

26

Combustion

2500

2400

Temperature (K)

2300

2200

2100

2000

1900 1.5 FIGURE 1.4

1.0

0.0 0.5 Enthalpy (kcal/g)

0.5

1.0

Equivalence ratio φ  1.0 values of Fig. 1.3 on an expanded scale.

molar specific heats than the triatomic molecules. For a given enthalpy content of reactants, the lower the mean specific heat of the product mixture, the greater the final flame temperature. At a given chemical enthalpy content of reactants, the larger the H/C ratio, the higher the temperature. This effect also comes about from the lower specific heat of water and its dissociation products compared to that of CO2 together with the higher endothermicity of CO2 dissociation. As one proceeds to more energetic reactants, the dissociation of

27

Chemical Thermodynamics and Flame Temperatures

2600

2500

Temperature (K)

2400

2300

2200

2100

2000

1900 1.5

1.0

0.0 0.5 Enthalpy (kcal/g)

0.5

1.0

FIGURE 1.5 Equivalence ratio φ  1.1 values of Fig. 1.3 on an expanded scale.

CO2 increases and the differences diminish. At the highest reaction enthalpies, the temperature for many fuels peaks not at the stoichiometric value, but, as stated, between φ  1.0 and 1.1 owing to lower mean specific heats of the richer products. At the highest temperatures and reaction enthalpies, the dissociation of the water is so complete that the system does not benefit from the heat of formation of the combustion product water. There is still a benefit from the heat or

28

Combustion

TABLE 1.2 Approximate Flame Temperatures Stoichiometric Mixtures, Initial Temperature 298 K Fuel

Oxidizer

Acetylene

of

Various

Pressure (atm)

Temperature (K)

Air

1

2600a

Acetylene

Oxygen

1

3410b

Carbon monoxide

Air

1

2400

Carbon monoxide

Oxygen

1

3220

Heptane

Air

1

2290

Heptane

Oxygen

1

3100

Hydrogen

Air

1

2400

Hydrogen

Oxygen

1

3080

Methane

Air

1

2210

Methane

Air

20

2270

Methane

Oxygen

1

3030

Methane

Oxygen

20

3460

a b

This maximum exists at φ  1.3. This maximum exists at φ  1.7.

formation of CO, the major dissociation product of CO2, so that the lower the H/C ratio, the higher the temperature. Thus for equivalence ratios around unity and very high energy content, the lower the H/C ratio, the greater the temperature; that is, the H/C curves intersect. As the pressure is increased in a combustion system, the amount of dissociation decreases and the temperature rises, as shown in Fig. 1.6. This observation follows directly from Le Chatelier’s principle. The effect is greatest, of course, at the stoichiometric air–fuel mixture ratio where the amount of dissociation is greatest. In a system that has little dissociation, the pressure effect on temperature is small. As one proceeds to a very lean operation, the temperatures and degree of dissociation are very low compared to the stoichiometric values; thus the temperature rise due to an increase in pressure is also very small. Figure 1.6 reports the calculated stoichiometric flame temperatures for propane and hydrogen in air and in pure oxygen as a function of pressure. Tables 1.3–1.6 list the product compositions of these fuels for three stoichiometries and pressures of 1 and 10 atm. As will be noted in Tables 1.3 and 1.5, the dissociation is minimal, amounting to about 3% at 1 atm and 2% at 10 atm. Thus one would not expect a large rise in temperature for this 10-fold increase

29

Chemical Thermodynamics and Flame Temperatures

T0  298 K, φ  1.0 3600

Adiabatic flame temperature (K)

3400

C3H8/O2 H2/O2

3200

3000

2800

2600 H2/Air 2400 C3H8/Air 2200

0

5

10

15

20

Pressure (atm) FIGURE 1.6 Calculated stoichiometric flame temperatures of propane and hydrogen in air and oxygen as a function of pressure.

TABLE 1.3 Equilibrium Combustion

Product

Composition

of

Propane–Air

φ

0.6

P (atm)

1

10

0

0

0.0125

0.0077

0.14041

0.1042

0.6

1.0 1

1.0 10

1.5 1

1.5 10

Species CO CO2

0.072

0.072

0.1027

0.1080

0.0494

0.0494

H

0

0

0.0004

0.0001

0.0003

0.0001

H2

0

0

0.0034

0.0019

0.0663

0.0664

H2O

0.096

0.096

0.1483

0.1512

0.1382

0.1383

NO

0.002

0.002

0.0023

0.0019

0

0

N2

0.751

0.751

0.7208

0.7237

0.6415

0.6416

O

0

0

0.0003

0.0001

0

0

OH

0.0003

0.0003

0.0033

0.0020

0.0001

0

O2

0.079

0.079

0.0059

0.0033

0

0

1974

1976

T(K) Dissociation (%)

1701

1702

2267

2318

3

2

30

Combustion

TABLE 1.4 Equilibrium Product Composition of Propane–Oxygen Combustion φ

0.6

P (atm)

1

0.6 10

1.0 1

1.0

1.5

10

1

1.5 10

Species CO

0.090

0.078

0.200

0.194

0.307

0.313

CO2

0.165

0.184

0.135

0.151

0.084

0.088

H

0.020

0.012

0.052

0.035

0.071

0.048

H2

0.023

0.016

0.063

0.056

0.154

0.155

H 2O

0.265

0.283

0.311

0.338

0.307

0.334

O

0.054

0.041

0.047

0.037

0.014

0.008

OH

0.089

0.089

0.095

0.098

0.051

0.046

O2

0.294

0.299

0.097

0.091

0.012

0.008

T(K) Dissociation (%)

2970

3236

3094

3411

27

23

55

51

3049

3331

TABLE 1.5 Equilibrium Product Composition of Hydrogen-Air Combustion φ

0.6

P (atm)

1

10

H

0

0

0.002

0

0.003

0.001

H2

0

0

0.015

0.009

0.147

0.148

0.6

1.0 1

1.0 10

1.5 1

1.5 10

Species

H2O

0.223

0.224

0.323

0.333

0.294

0.295

NO

0.003

0.003

0.003

0.002

0

0

N2

0.700

0.700

0.644

0.648

0.555

0.556

O

0

0

0.001

0

0

0

OH

0.001

0

0.007

0.004

0.001

0

O2 T(K) Dissociation (%)

0.073 1838

0.073 1840

0.005

0.003

2382

2442

3

2

0

0

2247

2252

31

Chemical Thermodynamics and Flame Temperatures

TABLE 1.6 Equilibrium Product Composition of Hydrogen–Oxygen Combustion φ

0.6

P (atm)

1

0.6 10

1.0 1

1.0 10

1.5 1

1.5 10

Species H

0.033

0.020

0.076

0.054

0.087

0.060

H2

0.052

0.040

0.152

0.141

0.309

0.318

H2O

0.554

0.580

0.580

0.627

0.535

0.568

O

0.047

0.035

0.033

0.025

0.009

0.005

OH

0.119

0.118

0.107

0.109

0.052

0.045

O2

0.205

0.207

0.052

0.044

0.007

0.004

T(K) Dissociation (%)

2980

3244

3077

3394

42

37

3003

3275

in pressure, as indeed Tables 1.3 and 1.5 and Fig. 1.7 reveal. This small variation is due mainly to the presence of large quantities of inert nitrogen. The results for pure oxygen (Tables 1.4 and 1.5) show a substantial degree of dissociation and about a 15% rise in temperature as the pressure increases from 1 to 10 atm. The effect of nitrogen as a diluent can be noted from Table 1.2, where the maximum flame temperatures of various fuels in air and pure oxygen are compared. Comparisons for methane in particular show very interesting effects. First, at 1 atm for pure oxygen the temperature rises about 37%; at 20 atm, over 50%. The rise in temperature for the methane–air system as the pressure is increased from 1 to 20 atm is only 2.7%, whereas for the oxygen system over the same pressure range the increase is about 14.2%. Again, these variations are due to the differences in the degree of dissociation. The dissociation for the equilibrium calculations is determined from the equilibrium constants of formation; moreover, from Le Chatelier’s principle, the higher the pressure the lower the amount of dissociation. Thus it is not surprising that a plot of ln Ptotal versus (1/Tf) gives mostly straight lines, as shown in Fig. 1.7. Recall that the equilibrium constant is equal to exp(ΔG°/RT). Many experimental systems in which nitrogen may undergo some reactions employ artificial air systems, replacing nitrogen with argon on a mole-for-mole basis. In this case the argon system creates much higher system temperatures because it absorbs much less of the heat of reaction owing to its lower specific

32

Combustion

Pressure as a function of inverse equilibrium temperatures of stoichiometric fuel-oxidizer systems initially at 298 K 100 Al/O2 C2H2/O2

Pressure (atm)

C2H2/Air CH4/O2

10

CH4/Air H2/O2 H2/Air 1

0.1 0.2

0.3

0.4

0.5

0.6

(1000 K)/T FIGURE 1.7 The variation of the stoichiometric flame temperature of various fuels in oxygen as a function of pressure in the for log P versus (1/Tf), where the initial system temperature is 298 K.

heat as a monotomic gas. The reverse is true, of course, when the nitrogen is replaced with a triatomic molecule such as carbon dioxide. Appendix B provides the adiabatic flame temperatures for stoichiometric mixtures of hydrocarbons and air for fuel molecules as large as C16 further illustrating the discussions of this section.

E. SUB- AND SUPER SONIC COMBUSTION THERMODYNAMICS 1. Comparisons In Chapter 4, Section G the concept of stabilizing a flame in a high-velocity stream is discussed. This discussion is related to streams that are subsonic. In essence what occurs is that the fuel is injected into a flowing stream and chemical reaction occurs in some type of flame zone. These types of chemically reacting streams are most obvious in air-breathing engines, particularly ramjets. In ramjets flying at supersonic speeds the air intake velocity must be lowered such that the flow velocity is subsonic entering the combustion chamber where the fuel is injected and the combustion stabilized by some flame holding technique. The inlet diffuser in this type of engine plays a very important role in the overall efficiency of the complete thrust generating process. Shock waves occur in the inlet when the vehicle is flying at supersonic speeds. Since there

Chemical Thermodynamics and Flame Temperatures

33

is a stagnation pressure loss with this diffuser process, the inlet to a ramjet must be carefully designed. At very high supersonic speeds the inlet stagnation pressure losses can be severe. It is the stagnation pressure at the inlet to the engine exhaust nozzle that determines an engine’s performance. Thus the concept of permitting complete supersonic flow through a properly designed converging–diverging inlet to enter the combustion chamber where the fuel must be injected, ignited, and stabilized requires a condition that the reaction heat release takes place in a reasonable combustion chamber length. In this case a converging–diverging section is still required to provide a thrust bearing surface. Whereas it will become evident in later chapters that normally the ignition time is much shorter than the time to complete combustion in a subsonic condition, in the supersonic case the reverse could be true. Thus there are three types of stagnation pressure losses in these subsonic and supersonic (scramjet) engines. They are due to inlet conditions, the stabilization process, and the combustion (heat release) process. As will be shown subsequently, although the inlet losses are smaller for the scramjet, stagnation pressure losses are greater in the supersonic combustion chamber. Stated in a general way, for a scramjet to be viable as a competitor to a subsonic ramjet, the scramjet must fly at very high Mach numbers where the inlet conditions for the subsonic case would cause large stagnation pressure losses. Even though inlet aerodynamics are outside the scope of this text, it is appropriate to establish in this chapter related to thermodynamics why subsonic combustion produces a lower stagnation pressure loss compared to supersonic combustion. This approach is possible since only the extent of heat release (enthalpy) and not the analysis of the reacting system is required. In a supersonic combustion chamber, a stabilization technique not causing losses and permitting rapid ignition still remain challenging endeavors.

2. Stagnation Pressure Considerations To understand the difference in stagnation pressure losses between subsonic and supersonic combustion one must consider sonic conditions in isoergic and isentropic flows; that is, one must deal with, as is done in fluid mechanics, the Fanno and Rayleigh lines. Following an early NACA report for these conditions, since the mass flow rate (ρuA) must remain constant, then for a constant area duct the momentum equation takes the form dP/ρ  u2 (du/u)  a 2 M 2 where a² is the speed of sound squared. M is the Mach number. For the condition of flow with variable area duct, in essence the equation simply becomes (1  M 2 ) (dP/ρ)  M 2 a 2 (dA/A)  u2 (dA/A)

(1.60)

34

Combustion

where A is the cross-sectional area of the flow chamber. For dA  0, which is the condition at the throat of a nozzle, it follows from Eq. (1.60) either dP  0 or M  1. Consequently, a minimum or maximum is reached, except when the sonic value is established in the throat of the nozzle. In this case the pressure gradient can be different. It follows then (dP/dA)  [1/ (1  M 2 )](ρ /A)

(1.61)

For M 1, (dP/dA) 0 and for M 1, (dP/dA) 0; that is, the pressure falls as one expands the area in supersonic flow and rises in subsonic flow. For adiabatic flow in a constant area duct, that is ρu  constant, one has for the Fanno line h  (u2 / 2)  h  where the lowercase h is the enthalpy per unit mass and the superscript o denotes the stagnation or total enthalpy. Considering P as a function of ρ and s (entropy), using Maxwell’s relations, the earlier definition of the sound speed and, for the approach here, the entropy as noted by Tds  dh  (dP/ρ) the expression of the Fanno line takes the form (dh/ds ) f  [ M 2 /(M 2  1)] T [1  (δ ln T /δ ln ρ )s ]

(1.62)

Since T varies in the same direction as ρ in an isotropic change, the term in brackets is positive. Thus for the flow conditions (dh/ds ) f 0 (dh/ds ) f 0 (dh/ds ) f  ∞

for M 1 for M 1 for M  1 (sonic condition)

where the subscript f denotes the Fanno condition. This derivation permits the Fanno line to be detailed in Fig. 1.8, which also contains the Rayleigh line to be discussed. The Rayleigh line is defined by the condition which results from heat exchange in a flow system and requires that the flow force remain constant, in essence for a constant area duct the condition can be written as dP  (ρu)du  0

35

Chemical Thermodynamics and Flame Temperatures

For a constant area duct note that (dP/ρ)  (du/u)  0 Again following the use of Eqs. (1.61) and (1.62), the development that ensues leads to [T /( M 2  1)]{[1  (δ ln T /δ ln ρ)s ] M 2  1}  (dh/ds)r

(1.63)

igh yle Ra <1 M g atin He

Fann o M<1

M1

A

h

M

>1

M g tin

>1

a

He

C

s FIGURE 1.8

An overlay of Fanno-Rayleigh conditions.

B

M1

where the subscript r denotes the Rayleigh condition. Examining Eq. (1.63), one will first note that at M  1, (dh/ds)r  , but also at some value of M 1 the term in the braces could equal 0 and thus (dh/ds)  0. Thus between this value and M  1, (dh/ds)r 0. At a value of M still less than the value for (dh/ds)r  0, the term in braces becomes negative and with the negative term multiple of the braces, (dh/ds)r becomes 0. These conditions determine the shape of the Rayleigh line, which is shown in Fig. 1.8 with the Fanno line. Since the conservation equations for a normal shock are represented by the Rayleigh and Fanno conditions, the final point must be on both lines and pass through the initial point. Since heat addition in a constant area duct cannot raise the velocity of the reacting fluid past the sonic speed, Fig. 1.8 represents the

36

Combustion

entropy change for both subsonic and supersonic flow for the same initial stagnation enthalpy. Equation (1.28) written in mass units can be represented as Δs  RΔ(ln P ) It is apparent the smaller change in entropy of subsonic combustion (A–B in Fig. 1.8) compared to supersonic combustion (C–B) establishes that there is a lower stagnation pressure loss in the subsonic case compared to the supersonic case. To repeat, for a scramjet to be viable, the inlet losses at the very high Mach number for subsonic combustion must be large enough to override its advantages gained in its energy release.z

PROBLEMS (Those with an asterisk require a numerical solution and use of an appropriate software program—See Appendix I.) 1. Suppose that methane and air in stoichiometric proportions are brought into a calorimeter at 500 K. The product composition is brought to the ambient temperature (298 K) by the cooling water. The pressure in the calorimeter is assumed to remain at 1 atm, but the water formed has condensed. Calculate the heat of reaction. 2. Calculate the flame temperature of normal octane (liquid) burning in air at an equivalence ratio of 0.5. For this problem assume there is no dissociation of the stable products formed. All reactants are at 298 K and the system operates at a pressure of 1 atm. Compare the results with those given by the graphs in the text. Explain any differences. 3. Carbon monoxide is oxidized to carbon dioxide in an excess of air (1 atm) in an afterburner so that the final temperature is 1300 K. Under the assumption of no dissociation, determine the air–fuel ratio required. Report the results on both a molar and mass basis. For the purposes of this problem assume that air has the composition of 1 mol of oxygen to 4 mol of nitrogen. The carbon monoxide and air enter the system at 298 K. 4. The exhaust of a carbureted automobile engine, which is operated slightly fuel-rich, has an efflux of unburned hydrocarbons entering the exhaust manifold. Assume that all the hydrocarbons are equivalent to ethylene (C2H4) and all the remaining gases are equivalent to inert nitrogen (N2). On a molar basis there are 40 mol of nitrogen for every mole of ethylene. The hydrocarbons are to be burned over an oxidative catalyst and converted to carbon dioxide and water vapor only. In order to accomplish this objective, ambient (298 K) air must be injected into the manifold before the catalyst. If the catalyst is to be maintained at 1000 K, how many moles of air per mole of ethylene must be added if the temperature of the manifold gases before air injection is 400 K and the composition of air is 1 mol of oxygen to 4 mol of nitrogen?

Chemical Thermodynamics and Flame Temperatures

37

5. A combustion test was performed at 20 atm in a hydrogen–oxygen system. Analysis of the combustion products, which were considered to be in equilibrium, revealed the following:

Compound

Mole fraction

H2O

0.493

H2

0.498

O2

0

O

0

H

0.020

OH

0.005

What was the combustion temperature in the test? 6. Whenever carbon monoxide is present in a reacting system, it is possible for it to disproportionate into carbon dioxide according to the equilibrium 2CO  Cs  CO2 Assume that such an equilibrium can exist in some crevice in an automotive cylinder or manifold. Determine whether raising the temperature decreases or increases the amount of carbon present. Determine the Kp for this equilibrium system and the effect of raising the pressure on the amount of carbon formed. 7. Determine the equilibrium constant Kp at 1000 K for the following reaction: 2CH 4  2H 2  C2 H 4 8. The atmosphere of Venus is said to contain 5% carbon dioxide and 95% nitrogen by volume. It is possible to simulate this atmosphere for Venus reentry studies by burning gaseous cyanogen (C2N2) and oxygen and diluting with nitrogen in the stagnation chamber of a continuously operating wind tunnel. If the stagnation pressure is 20 atm, what is the maximum stagnation temperature that could be reached while maintaining Venus atmosphere conditions? If the stagnation pressure were 1 atm, what would the maximum temperature be? Assume all gases enter the chamber at 298 K. Take the heat of formation of cyanogen as (ΔH f )298  374 kJ/mol. 9. A mixture of 1 mol of N2 and 0.5 mol O2 is heated to 4000 K at 0.5 atm, affording an equilibrium mixture of N2, O2, and NO only. If the O2 and N2 were initially at 298 K and the process is one of steady heating, how

38

Combustion

much heat is required to bring the final mixture to 4000 K on the basis of one initial mole at N2? 10. Calculate the adiabatic decomposition temperature of benzene under the constant pressure condition of 20 atm. Assume that benzene enters the decomposition chamber in the liquid state at 298 K and decomposes into the following products: carbon (graphite), hydrogen, and methane. 11. Calculate the flame temperature and product composition of liquid ethylene oxide decomposing at 20 atm by the irreversible reaction C2 H 4 O (liq) → aCO  bCH 4  CH 2  dC2 H 4 The four products are as specified. The equilibrium known to exist is 2CH 4  2H 2  C2 H 4 The heat of formation of liquid ethylene oxide is ΔH f , 298  76.7 kJ / mol It enters the decomposition chamber at 298 K. 12. Liquid hydrazine (N2H4) decomposes exothermically in a monopropellant rocket operating at 100 atm chamber pressure. The products formed in the chamber are N2, H2, and ammonia (NH3) according to the irreversible reaction N 2 H 4 (liq ) → aN 2  bH 2  cNH3 Determine the adiabatic decomposition temperature and the product composition a, b, and c. Take the standard heat of formation of liquid hydrazine as 50.07 kJ/mol. The hydrazine enters the system at 298 K. 13. Gaseous hydrogen and oxygen are burned at 1 atm under the rich conditions designated by the following combustion reaction: O2  5H 2 → aH 2 O  bH 2  cH The gases enter at 298 K. Calculate the adiabatic flame temperature and the product composition a, b, and c. 14. The liquid propellant rocket combination nitrogen tetroxide (N2O4) and UDMH (unsymmetrical dimethyl hydrazine) has optimum performance at an oxidizer-to-fuel weight ratio of 2 at a chamber pressure of 67 atm. Assume that the products of combustion of this mixture are N2, CO2, H2O, CO, H2, O, H, OH, and NO. Set down the equations necessary to calculate the adiabatic combustion temperature and the actual product composition under these conditions. These equations should contain all the numerical

Chemical Thermodynamics and Flame Temperatures

39

data in the description of the problem and in the tables in the appendices. The heats of formation of the reactants are N 2 O 4 (liq ), UDMH (liq),

ΔH f,298  2.1 kJ/mol ΔH f,298  53.2 kJ/mol

The propellants enter the combustion chamber at 298 K. 15. Consider a fuel burning in inert airs and oxygen where the combustion requirement is only 0.21 moles of oxygen. Order the following mixtures as to their adiabatic flame temperatures with the given fuel. a) pure O2 b) 0.21 O2  0.79 N2 (air) c) 0.21 O2  0.79 Ar d) 0.21 O2  0.79 CO2 16. Propellant chemists have proposed a new high energy liquid oxidizer, penta-oxygen O5, which is also a monopropellant. Calculate the monopropellant decomposition temperature at a chamber pressure of 10 atm if it assumed the only products are O atoms and O2 molecules. The heat of formation of the new oxidizer is estimated to be very high, 1025 kJ/mol. Obviously, the amounts of O2 and O must be calculated for one mole of O5 decomposing. The O5 enters the system at 298 K. Hint: The answer will lie somewhere between 4000 and 5000 K. 17.*Determine the amount of CO2 and H2O dissociation in a mixture initially consisting of 1 mol of CO2, 2 mol of H2O, and 7.5 mol of N2 at temperatures of 1000, 2000, and 3000 K at atmospheric pressure and 50 atm. Use a numerical program such as the NASA Glenn Chemical Equilibrium Analysis (CEA) program or one included with CHEMKIN (Equil for CHEMKIN II and III, Equilibrium-Gas for CHEMKIN IV). Appendix I provides information on several of these programs. 18.*Calculate the constant pressure adiabatic flame temperature and equilibrium composition of stoichiometric mixtures of methane (CH4) and gaseous methanol (CH3OH) with air initially at 300 K and 1 atm. Perform the calculation first with the equilibrium program included with CHEMKIN (see Appendix I), and then compare your results to those obtained with the CEA program (see Appendix I). Compare the mixture compositions and flame temperatures and discuss the trends. 19.*Calculate the adiabatic flame temperature of a stoichiometric methane–air mixture for a constant pressure process. Compare and discuss this temperature to those obtained when the N2 in the air has been replaced with He, Ar, and CO2. Assume the mixture is initially at 300 K and 1 atm. 20.*Consider a carbon monoxide and air mixture undergoing constant pressure, adiabatic combustion. Determine the adiabatic flame temperature and equilibrium mixture composition for mixtures with equivalence ratios varying from 0.5 to 3 in increments of 0.5. Plot the temperature and

40

Combustion

concentrations of CO, CO2, O2, O, and N2 as a function of equivalence ratio. Repeat the calculations with a 33% CO/67% H2 (by volume) fuel mixture. Plot the temperature and concentrations of CO, CO2, O2, O, N2, H2, H2O, OH, and H. Compare the two systems and discuss the trends of temperature and species concentration. 21.*A Diesel engine with a compression ratio of 20:1 operates on liquid decane (C10H22) with an overall air/fuel (massair/massfuel) mixture ratio of 18:1. Assuming isentropic compression of air initially at 298 K and 1 atm followed by fuel injection and combustion at constant pressure, determine the equilibrium flame temperature and mixture composition. Include NO and NO2 in your equilibrium calculations. In the NASA CEA program, equilibrium calculations for fuels, which will not exist at equilibrium conditions, can be performed with knowledge of only the chemical formula and the heat of formation. The heat of formation of liquid decane is 301.04 kJ/mol. In other programs, such as CHEMKIN, thermodynamic data are required for the heat of formation, entropy, and specific heat as a function of temperature. Such data are often represented as polynominals to describe the temperature dependence. In the CHEMKIN thermodynamic database (adopted from the NASA Chemical Equilibrium code, Gordon, S., and McBride, B. J., NASA Report SP-273, 1971), the specific heat, enthalpy, and entropy are represented by the following expressions. C p /Ru  a1  a2T  a3T 2  a4T 3  a5T 4 H  /RuT  a1  a2T/ 2  a3T 2 / 3  a4T 3 / 4  a5T 4 / 5  a6 /T S  /Ru  a1 ln T  a2T  a3T 2 / 2  a4T 3 / 4  a5T 4 / 5  a7 The coefficients a1 through a7 are generally provided for both high-and low-temperature ranges. Thermodynamic data in CHEMKIN format for liquid decane is given below. C10H22(L) B01/00C 10.H 22. 0. 0.L 298.150 446.830 C 142.28468 0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00 3.77595368E+01 5.43284903E-04-1.44050795E-06 1.25634293E-09 0.00000000E+00-4.74783720E+04-1.64025285E02-3.62064632E+04

1 2 3 4

In the data above, the first line provides the chemical name, a comment, the elemental composition, the phase, and the temperature range over which the data are reported. In lines 2 through 4, the high-temperature coefficients a1,…, a7 are presented first followed by the low temperature coefficients. For more information, refer to Kee et al. (Kee, R. J., Rupley, F. M., and Miller, J. A., “The Chemkin Thermodynamic Database,” Sandia Report, SAND87-8215B, reprinted March 1991). 22.*A spark ignition engine with a 8:1 compression ratio is tested with a reference fuel mixture consisting of 87% iso-octane (i-C8H18) and

Chemical Thermodynamics and Flame Temperatures

41

13% normal-heptane (n-C7H16) by volume. Assuming combustion takes place at constant volume with air that has been compressed isentropically, calculate the equilibrium flame temperature and mixture composition for a stoichiometric mixture. Include NO and NO2 in your equilibrium calculation. The heats of formation of iso-octane and n-heptane are 250.26 and 224.35 kJ/mol, respectively. If necessary, the thermodynamic data in CHEMKIN format are provided below. See question 19 for a description of these values. C7H16(L) n-hept P10/75C 7.H 16. 0. 0.C 182.580 380.000 C 100.20194 0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00 6.98058594E+01-6.30275879E-01 3.08862295E-03 -6.40121661E-06 5.09570496E-09-3.68238127E+04-2.61086466E+02-2.69829491E+04

1 2 3 4

C8H18(L) isooct L10/82C 8.H 18. 0. 0.C 165.790 380.000 C 114.22852. 0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00 1.75199280E+01 1.57483711E-02 7.35946809E-05 -6.10398277E-10 4.70619213E-13-3.77423257E+04-6.83211023E+01-3.11696059E+04

1 2 3 4

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

10.

JANAF Thermochemical Tables, 3rd Ed., J. Phys. Chem. Ref Data 14 (1985). Nat. Bur. Stand. (U.S.), Circ. No. C461 (1947). Glassman, I., and Scott, J. E. Jr., Jet Propulsion 24, 95 (1954). Gordon, S., and McBride, B. J., NASA (Spec. Publ.), SP NASA SP-272, Int. Rev. (Mar. 1976). Eberstein, I. J., and Glassman, I., Prog. Astronaut. Rocketry 2, 95 (1954). Huff, V. N. and Morell, J. E., Natl. Advis. Comm. Aeronaut. Tech Note NASA TN-1133 (1950). Glassman, I., and Sawyer, R. F., “The Performance of Chemical Propellants.” Chap. I 1, Technivision, London, 1970. Reynolds, W. C., “Stanjan,” Dept. Mech. Eng. Stanford University, Stanford, CA, 1986. Kee, R. J., Rupley, F. M., and Miller, J. A., CHEMKIN-II: “A FORTRAN Chemical Kinetics Package for the Analysis of Gas-Phase Chemical Kinetics,” Sandia National Laboratories, Livermore, CA, Sandia Report SAWD89-8009, 1989. Glassman, I., and Clark, G., East. States Combust. Inst. Meet., Providence, RI, Pap. No. 12 (Nov. 1983).

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Chapter 2

Chemical Kinetics A. INTRODUCTION Flames will propagate through only those chemical mixtures that are capable of reacting quickly enough to be considered explosive. Indeed, the expression “explosive” essentially specifies very rapid reaction. From the standpoint of combustion, the interest in chemical kinetic phenomena has generally focused on the conditions under which chemical systems undergo explosive reaction. Recently, however, great interest has developed in the rates and mechanisms of steady (nonexplosive) chemical reactions, since most of the known complex pollutants form in zones of steady, usually lower-temperature, reactions during, and even after, the combustion process. These essential features of chemical kinetics, which occur frequently in combustion phenomena, are reviewed in this chapter. For a more detailed understanding of any of these aspects and a thorough coverage of the subject, refer to any of the books on chemical kinetics, such as those listed in Refs. [1, 1a].

B. RATES OF REACTIONS AND THEIR TEMPERATURE DEPENDENCE All chemical reactions, whether of the hydrolysis, acid–base, or combustion type, take place at a definite rate and depend on the conditions of the system. The most important of these conditions are the concentration of the reactants, the temperature, radiation effects, and the presence of a catalyst or inhibitor. The rate of the reaction may be expressed in terms of the concentration of any of the reacting substances or of any reaction product; that is, the rate may be expressed as the rate of decrease of the concentration of a reactant or the rate of increase of a reaction product. A stoichiometric relation describing a one-step chemical reaction of arbitrary complexity can be represented by the equation [2.2, 2.3] n

n

j 1

j 1

∑ ν j(M j )  ∑ ν j (M j )

(2.1)

where νj is the stoichiometric coefficient of the reactants, ν j is the stoichiometric coefficient of the products, M is an arbitrary specification of all chemical 43

44

Combustion

species, and n is the total number of species involved. If a species represented by Mj does not occur as a reactant or product, its νj equals zero. Consider, as an example, the recombination of H atoms in the presence of H atoms, that is, the reaction H  H  H → H2  H M1  H, M2  H 2 n  2, ν 1  3, ν 1  1, ν 2  1 The reason for following this complex notation will become apparent shortly. The law of mass action, which is confirmed experimentally, states that the rate of disappearance of a chemical species i, defined as RRi, is proportional to the product of the concentrations of the reacting chemical species, where each concentration is raised to a power equal to the corresponding stoichiometric coefficient; that is,

RRi ∼

n

∏ (M j )ν , j

j 1

n

RRi  k ∏ (M j )ν j

(2.2)

j 1

where k is the proportionality constant called the specific reaction rate coefficient. In Eq. (2.2) νj is also given the symbol n, which is called the overall order of the reaction; νj itself would be the order of the reaction with respect to species j. In an actual reacting system, the rate of change of the concentration of a given species i is given by n d ( Mi ) ⎡  ⎢⎣ ν i  ν i ⎤⎦⎥ RR  ⎡⎣⎢ ν i  ν i ⎤⎦⎥ k ∏ (M j )ν j dt j 1

(2.3)

since ν j moles of Mi are formed for every νj moles of Mi consumed. For the previous example, then, d(H)/dt  2k(H)3. The use of this complex representation prevents error in sign and eliminates confusion when stoichiometric coefficients are different from 1. In many systems Mj can be formed not only from a single-step reaction such as that represented by Eq. (2.3), but also from many different such steps, leading to a rather complex formulation of the overall rate. However, for a singlestep reaction such as Eq. (2.3), νj not only represents the overall order of the reaction, but also the molecularity, which is defined as the number of molecules that interact in the reaction step. Generally the molecularity of most reactions of interest will be 2 or 3. For a complex reaction scheme, the concept of molecularity is not appropriate and the overall order can take various values including fractional ones.

45

Chemical Kinetics

1. The Arrhenius Rate Expression In most chemical reactions the rates are dominated by collisions of two species that may have the capability to react. Thus, most simple reactions are secondorder. Other reactions are dominated by a loose bond-breaking step and thus are first-order. Most of these latter type reactions fall in the class of decomposition processes. Isomerization reactions are also found to be first-order. According to Lindemann’s theory [1, 4] of first-order processes, first-order reactions occur as a result of a two-step process. This point will be discussed in a subsequent section. An arbitrary second-order reaction may be written as A  B→ C  D

(2.4)

where a real example would be the reaction of oxygen atoms with nitrogen molecules O  N 2 → NO  N For the arbitrary reaction (2.4), the rate expression takes the form RR 

d (A) d (B) d (C) d (D)  k (A)(B)    dt dt dt dt

(2.5)

The convention used throughout this book is that parentheses around a chemical symbol signify the concentration of that species in moles or mass per cubic centimeter. Specifying the reaction in this manner does not infer that every collision of the reactants A and B would lead to products or cause the disappearance of either reactant. Arrhenius [5] put forth a simple theory that accounts for this fact and gives a temperature dependence of k. According to Arrhenius, only molecules that possess energy greater than a certain amount, E, will react. Molecules acquire the additional energy necessary from collisions induced by the thermal condition that exists. These high-energy activated molecules lead to products. Arrhenius’ postulate may be written as RR  Z AB exp(E/RT )

(2.6)

where ZAB is the gas kinetic collision frequency and exp(E/RT) is the Boltzmann factor. Kinetic theory shows that the Boltzmann factor gives the fraction of all collisions that have an energy greater than E. The energy term in the Boltzmann factor may be considered as the size of the barrier along a potential energy surface for a system of reactants going to products, as shown schematically in Fig. 2.1. The state of the reacting species at this activated energy can be regarded as some intermediate complex that

46

Combustion

Energy

EA  Ef, Eb Ef Reactants

Eb

H

Products Reaction coordinate FIGURE 2.1

Energy as a function of a reaction coordinate for a reacting system.

leads to the products. This energy is referred to as the activation energy of the reaction and is generally given the symbol EA. In Fig. 2.1, this energy is given the symbol Ef, to distinguish it from the condition in which the product species can revert to reactants by a backward reaction. The activation energy of this backward reaction is represented by Eb and is obviously much larger than Ef for the forward step. Figure 2.1 shows an exothermic condition for reactants going to products. The relationship between the activation energy and the heat of reaction has been developed [1a]. Generally, the more exothermic a reaction is, the smaller the activation energy. In complex systems, the energy release from one such reaction can sustain other, endothermic reactions, such as that represented in Fig. 2.1 for products reverting back to reactants. For example, once the reaction is initiated, acetylene will decompose to the elements in a monopropellant rocket in a sustained fashion because the energy release of the decomposition process is greater than the activation energy of the process. In contrast, a calculation of the decomposition of benzene shows the process to be exothermic, but the activation energy of the benzene decomposition process is so large that it will not sustain monopropellant decomposition. For this reason, acetylene is considered an unstable species and benzene a stable one. Considering again Eq. (2.6) and referring to E as an activation energy, attention is focused on the collision rate ZAB, which from simple kinetic theory can be represented by 2 [8π k T/μ] Z AB  (A)(B) σAB B

1/ 2

(2.7)

where σAB is the hard sphere collision diameter, kB the Boltzmann constant, μ is the reduced mass [mAmB/(mA  mB)], and m is the mass of the species. ZAB may be written in the form Z AB  Z AB (A)(B)

(2.7a)

47

Chemical Kinetics

2 [8π k T/μ ]1/ 2. Thus, the Arrhenius form for the rate is where Z AB  σAB B

RR  Z AB (A)(B) exp(E/RT ) When one compares this to the reaction rate written from the law of mass action [Eq. (2.2)], one finds that k  Z AB exp (E/RT )  Z ABT 1/ 2 exp(E/RT )

(2.8)

Thus, the important conclusion is that the specific reaction rate constant k is dependent on temperature alone and is independent of concentration. Actually, when complex molecules are reacting, not every collision has the proper steric orientation for the specific reaction to take place. To include the steric probability, one writes k as k  Z ABT 1/ 2 [exp(E/RT )]℘

(2.9)

where  is a steric factor, which can be a very small number at times. Most generally, however, the Arrhenius form of the reaction rate constant is written as k  const T 1/ 2 exp(E/RT )  A exp(E/RT )

(2.10)

where the constant A takes into account the steric factor and the terms in the collision frequency other than the concentrations and is referred to as the kinetic pre-exponential A factor. The factor A as represented in Eq. (2.10) has a very mild T1/2 temperature dependence that is generally ignored when plotting data. The form of Eq. (2.10) holds well for many reactions, showing an increase in k with T that permits convenient straight-line correlations of data on ln k versus (1/T) plots. Data that correlate as a straight line on a ln k versus (1/T) plot are said to follow Arrhenius kinetics, and plots of the logarithm of rates or rate constants as a function of (1/T) are referred to as Arrhenius plots. The slopes of lines on these plots are equal to (E/R); thus the activation energy may be determined readily (see Fig. 2.2). Low activation energy processes proceed faster than high activation energy processes at low temperatures and are much less temperature-sensitive. At high temperatures, high activation energy reactions can prevail because of this temperature sensitivity.

2. Transition State and Recombination Rate Theories There are two classes of reactions for which Eq. (2.10) is not suitable. Recombination reactions and low activation energy free-radical reactions in which the temperature dependence in the pre-exponential term assumes more importance. In this low-activation, free-radical case the approach known as

48

Combustion

In k

Slope  EA/R

1/T FIGURE 2.2 temperature.

Arrhenius plot of the specific reaction rate constant as a function of the reciprocal

absolute or transition state theory of reaction rates gives a more appropriate correlation of reaction rate data with temperature. In this theory the reactants are assumed to be in equilibrium with an activated complex. One of the vibrational modes in the complex is considered loose and permits the complex to dissociate to products. Fig. 2.1 is again an appropriate representation, where the reactants are in equilibrium with an activated complex, which is shown by the curve peak along the extent of the reaction coordinate. When the equilibrium constant for this situation is written in terms of partition functions and if the frequency of the loose vibration is allowed to approach zero, a rate constant can be derived in the following fashion. The concentration of the activated complex may be calculated by statistical thermodynamics in terms of the reactant concentrations and an equilibrium constant [1, 6]. If the reaction scheme is written as A  BC  (ABC)# → AB  C

(2.11)

the equilibrium constant with respect to the reactants may be written as K# 

(ABC)# (A)(BC)

(2.12)

where the symbol # refers to the activated complex. As discussed in Chapter 1, since K# is expressed in terms of concentration, it is pressure-dependent. Statistical thermodynamics relates equilibrium constants to partition functions; thus for the case in question, one finds [6] K# 

⎛ E ⎞⎟ (QT )# exp ⎜⎜ ⎟ ⎜⎝ RT ⎟⎠ (QT )A (QT )BC

(2.13)

49

Chemical Kinetics

where QT is the total partition function of each species in the reaction. QT can be considered separable into vibrational, rotational, and translation partition functions. However, one of the terms in the vibrational partition function part of Q# is different in character from the rest because it corresponds to a very loose vibration that allows the complex to dissociate into products. The complete vibrational partition function is written as 1 Qvib  ∏ ⎡⎣1  exp(hvi /kBT ) ⎤⎦

(2.14)

i

where h is Planck’s constant and vi is the vibrational frequency of the ith mode. For the loose vibration, one term of the complete vibrational partition function can be separated and its value employed when ν tends to zero, lim [1  exp(hv/kBT )]1  (kBT/hv)

(2.15)

{(ABC)# /[(A)(BC)]}  {[(QT1 )# (kBT/hv)]/[(QT )A (QT )BC ]}  exp (E/RT )

(2.16)

v→0

Thus

which rearranges to ν(ABC)#  {[(A)(BC)(kBT/h)(QT1 )# ]/[(QT )A (QT )BC ]}  exp(E/RT )

(2.17)

where (QT1)# is the partition function of the activated complex evaluated for all vibrational frequencies except the loose one. The term v(ABC)# on the lefthand side of Eq. (2.17) is the frequency of the activated complex in the degree of freedom corresponding to its decomposition mode and is therefore the frequency of decomposition. Thus, k  (kBT/h)[(QT1 )# /(QT )A (QT )BC ] exp(EA /RT )

(2.18)

is the expression for the specific reaction rate as derived from transition state theory. If species A is only a diatomic molecule, the reaction scheme can be represented by A  A # → products

(2.19)

50

Combustion

Thus (QT1)# goes to 1. There is only one bond in A, so Qvib, A  [1  exp(hvA /kBT )]1

(2.20)

k  (kBT/h)[1  exp(hvA /kBT )] exp(E/RT )

(2.21)

Then

Normally in decomposition systems, vA of the stable molecule is large, then the term in square brackets goes to 1 and k  (kBT/h ) exp (E/RT )

(2.22)

Note that the term (kBT/h) gives a general order of the pre-exponential term for these dissociation processes. Although the rate constant will increase monotonically with T for Arrhenius’ collision theory, examination of Eqs. (2.18) and (2.22) reveals that a nonmonotonic trend can be found [7] for the low activation energy processes represented by transition state theory. Thus, data that show curvature on an Arrhenius plot probably represent a reacting system in which an intermediate complex forms and in which the activation energy is low. As the results from Problem 1 of this chapter reveal, the term (kBT/h) and the Arrhenius pre-exponential term given by Eq. (2.7a) are approximately the same and/or about 1014 cm3 mol1s1 at 1000 K. This agreement is true when there is little entropy change between the reactants and the transition state and is nearly true for most cases. Thus one should generally expect pre-exponential values to fall in a range near 1013–1014 cm3 mol1s1. When quantities far different from this range are reported, one should conclude that the representative expression is an empirical fit to some experimental data over a limited temperature range. The earliest representation of an important combustion reaction that showed curvature on an Arrhenius plot was for the CO  OH reaction as given in Ref. [7], which, by application of transition state theory, correlated a wide temperature range of experimental data. Since then, consideration of transition state theory has been given to many other reactions important to combustion [8]. The use of transition state theory as a convenient expression of rate data is obviously complex owing to the presence of the temperature-dependent partition functions. Most researchers working in the area of chemical kinetic modeling have found it necessary to adopt a uniform means of expressing the temperature variation of rate data and consequently have adopted a modified Arrhenius form k  AT n exp(E/RT )

(2.23)

where the power of T accounts for all the pre-exponential temperaturedependent terms in Eqs. (2.10), (2.18), and (2.22). Since most elementary

51

Chemical Kinetics

binary reactions exhibit Arrhenius behavior over modest ranges of temperature, the temperature dependence can usually be incorporated with sufficient accuracy into the exponential alone; thus, for most data n  0 is adequate, as will be noted for the extensive listing in the appendixes. However, for the large temperature ranges found in combustion, “non-Arrhenius” behavior of rate constants tends to be the rule rather than the exception, particularly for processes that have a small energy barrier. It should be noted that for these processes the pre-exponential factor that contains the ratio of partition functions (which are weak functions of temperature compared to an exponential) corresponds roughly to a T n dependence with n in the 1–2 range [9]. Indeed the values of n for the rate data expressions reported in the appendixes fall within this range. Generally the values of n listed apply only over a limited range of temperatures and they may be evaluated by the techniques of thermochemical kinetics [10]. The units for the reaction rate constant k when the reaction is of order n (different from the n power of T) will be [(conc)n1 (time)]1. Thus, for a firstorder reaction the units of k are in reciprocal seconds (s1), and for a secondorder reaction process the units are in cubic centimeter per mol per second ( cm3 mol1 s1). Thus, as shown in Appendix C, the most commonly used units for kinetic rates are cm3 mol kJ, where kilojoules are used for the activation energy. Radical recombination is another class of reactions in which the Arrhenius expression will not hold. When simple radicals recombine to form a product, the energy liberated in the process is sufficiently great to cause the product to decompose into the original radicals. Ordinarily, a third body is necessary to remove this energy to complete the recombination. If the molecule formed in a recombination process has a large number of internal (generally vibrational) degrees of freedom, it can redistribute the energy of formation among these degrees, so a third body is not necessary. In some cases the recombination process can be stabilized if the formed molecule dissipates some energy radiatively (chemiluminescence) or collides with a surface and dissipates energy in this manner. If one follows the approach of Landau and Teller [11], who in dealing with vibrational relaxation developed an expression by averaging a transition probability based on the relative molecular velocity over the Maxwellian distribution, one can obtain the following expression for the recombination rate constant [6]: k ∼ exp(C/T )1/ 3

(2.24)

where C is a positive constant that depends on the physical properties of the species of concern [6]. Thus, for radical recombination, the reaction rate constant decreases mildly with the temperature, as one would intuitively expect. In dealing with the recombination of radicals in nozzle flow, one should keep this mild

52

Combustion

temperature dependence in mind. Recall the example of H atom recombination given earlier. If one writes M as any (or all) third body in the system, the equation takes the form H  H  M → H2  M

(2.25)

The rate of formation of H2 is third-order and given by d (H 2 )/dt  k (H)2 (M)

(2.25a)

Thus, in expanding dissociated gases through a nozzle, the velocity increases and the temperature and pressure decrease. The rate constant for this process thus increases, but only slightly. The pressure affects the concentrations and since the reaction is third-order, it enters the rate as a cubed term. In all, then, the rate of recombination in the high-velocity expanding region decreases owing to the pressure term. The point to be made is that third-body recombination reactions are mostly pressure-sensitive, generally favored at higher pressure, and rarely occur at very low pressures.

C. SIMULTANEOUS INTERDEPENDENT REACTIONS In complex reacting systems, such as those in combustion processes, a simple one-step rate expression will not suffice. Generally, one finds simultaneous, interdependent reactions or chain reactions. The most frequently occurring simultaneous, interdependent reaction mechanism is the case in which the product, as its concentration is increased, begins to dissociate into the reactants. The classical example is the hydrogen–iodine reaction: k

f ⎯⎯⎯ ⎯⎯ → 2HI H2  I2 ← ⎯

kb

(2.26)

The rate of formation of HI is then affected by two rate constants, kf and kb, and is written as d (HI)/dt  2 kf (H 2 ) (I 2 )  2 kb (HI)2

(2.27)

in which the numerical value 2 should be noted. At equilibrium, the rate of formation of HI is zero, and one finds that 2 0 2 kf (H 2 )eq (I 2 )eq  2 kb (HI)eq

(2.28)

53

Chemical Kinetics

where the subscript eq designates the equilibrium concentrations. Thus, 2 (HI)eq kf  ≡ Kc kb (H 2 )eq (I 2 )eq

(2.29)

that is, the forward and backward rate constants are related to the equilibrium constant K based on concentrations (Kc). The equilibrium constants are calculated from basic thermodynamic principles as discussed in Section 1C, and the relationship (kf / kb)  Kc holds for any reacting system. The calculation of the equilibrium constant is much more accurate than experimental measurements of specific reaction rate constants. Thus, given a measurement of a specific reaction rate constant, the reverse rate constant is determined from the relationship Kc  (kf / kb). For the particular reaction in Eq. (2.29), Kc is not pressure-dependent as there is a concentration squared in both the numerator and denominator. Indeed, Kc equals (kf/ kb)  Kp only when the concentration powers cancel. With this equilibrium consideration the rate expression for the formation of HI becomes k d (HI)  2 kf (H 2 )(I 2 )  2 f (HI)2 dt Kc

(2.30)

which shows there is only one independent rate constant in the problem.

D. CHAIN REACTIONS In most instances, two reacting molecules do not react directly as H2 and I2 do; rather one molecule dissociates first to form radicals. These radicals then initiate a chain of steps. Interestingly, this procedure occurs in the reaction of H2 with another halogen, Br2. Experimentally, Bodenstein [12] found that the rate of formation of HBr obeys the expression kexp (H 2 )(Br2 )1/ 2 d (HBr)  dt 1  k exp [(HBr)/(Br2 )] This expression shows that HBr is inhibiting to its own formation.

(2.31)

54

Combustion

Bodenstein explained this result by suggesting that the H2ßBr2 reaction was chain in character and initiated by a radical (Bri) formed by the thermal dissociation of Br2. He proposed the following steps: k

1 → 2 Br i  M (1) M  Br2 ⎯ ⎯⎯

}

Chain initiating step

k → HBr  Hi ⎫⎪⎪ (2) Br i  H 2 ⎯ ⎯2 ⎯ ⎪⎪ k3 (3) Hi  Br2 ⎯ ⎯⎯ → HBr  Br i ⎬ Chain carrying or propagating steps ⎪ k4 (4) Hi  HBr ⎯ ⎯⎯ → H 2  Br i ⎪⎪⎪ ⎪⎭ k Chain terminating step (5) M  2 Br i ⎯ ⎯5 ⎯ → Br2  M

}

The Br2 bond energy is approximately 189 kJ/mol and the H2 bond energy is approximately 427 kJ/mol. Consequently, except for the very highest temperature, Br2 dissociation will be the initiating step. These dissociation steps follow Arrhenius kinetics and form a plot similar to that shown in Fig. 2.2. In Fig. 2.3 two Arrhenius plots are shown, one for a high activation energy step and another for a low activation energy step. One can readily observe that for low temperature, the smaller EA step prevails. Perhaps the most important of the various chain types is the chain step that is necessary to achieve nonthermal explosions. This chain step, called chain branching, is one in which two radicals are created for each radical consumed. Two typical chain branching steps that occur in the H2ßO2 reaction system are Hi  O2 → iOH  iOi iOi  H 2 → iOH  Hi where the dot next to or over a species is the convention for designating a radical. Such branching will usually occur when the monoradical (such as H•) formed by breaking a single bond reacts with a species containing a double bond type structure (such as that in O2) or when a biradical (such as •O•) formed by breaking a double bond reacts with a saturated molecule (such as H2 or RH where R is any organic radical). For an extensive discussion of chain reactions, refer to the monograph by Dainton [13]. As shown in the H2ßBr2 example, radicals are produced by dissociation of a reactant in the initiation process. These types of dissociation reactions are highly endothermic and therefore quite slow. The activation energy of these processes would be in the range of 160–460 kJ/mol. Propagation reactions similar to reactions (2.2)–(2.4) in the H2ßBr2 example are important because they determine the rate at which the chain continues. For most propagation reactions of importance in combustion, activation energies normally lie between 0 and 40 kJ/mol. Obviously, branching chain steps are a special case of propagating steps and, as mentioned, these are the steps that lead to explosion.

55

Chemical Kinetics

4

In k

I

3 II 2 1

1/T FIGURE 2.3 Plot of ln k versus 1/T. Region I denotes a high activation energy process and Region II a low activation energy process. Numerals designate conditions to be discussed in Chapter 3.

Branching steps need not necessarily occur rapidly because of the multiplication effect; thus, their activation energies may be higher than those of the linear propagation reactions with which they compete [14]. Termination occurs when two radicals recombine; they need not be similar to those shown in the H2ßBr2 case. Termination can also occur when a radical reacts with a molecule to give either a molecular species or a radical of lower activity that cannot propagate a chain. Since recombination processes are exothermic, the energy developed must be removed by another source, as discussed previously. The source can be another gaseous molecule M, as shown in the example, or a wall. For the gaseous case, a termolecular or third-order reaction is required; consequently, these reactions are slower than other types except at high pressures. In writing chain mechanisms note that backward reactions are often written as an individual step; that is, reaction (2.4) of the H2ßBr2 scheme is the backward step of reaction (2.2). The inverse of reaction (2.3) proceeds very slowly; it is therefore not important in the system and is usually omitted for the H2ßBr2 example. From the five chain steps written for the H2ßBr2 reaction, one can write an expression for the HBr formation rate: d (HBr)  k2 (Br)(H 2 )  k3 (H)(Br2 )  k4 (H)(HBr) dt

(2.32)

In experimental systems, it is usually very difficult to measure the concentration of the radicals that are important intermediates. However, one would

56

Combustion

like to be able to relate the radical concentrations to other known or measurable quantities. It is possible to achieve this objective by the so-called steadystate approximation for the reaction’s radical intermediates. The assumption is that the radicals form and react very rapidly so that the radical concentration changes only very slightly with time, thereby approximating a steady-state concentration. Thus, one writes the equations for the rate of change of the radical concentration, then sets them equal to zero. For the H2ßBr2 system, then, one has for (H) and (Br) d (H)  k2 (Br)(H 2 )  k3 (H)(Br2 )  k4 (H)(HBr) ≅ 0 dt d (Br)  2 k1 (Br2 )(M)  k2 (Br)(H 2 )  k3 (H)(Br2 ) dt  k4 (H)(HBr)  2 k5 (Br)2 (M) ≅ 0

(2.33)

(2.34)

Writing these two equations equal to zero does not imply that equilibrium conditions exist, as was the case for Eq. (2.28). It is also important to realize that the steady-state approximation does not imply that the rate of change of the radical concentration is necessarily zero, but rather that the rate terms for the expressions of radical formation and disappearance are much greater than the radical concentration rate term. That is, the sum of the positive terms and the sum of the negative terms on the right-hand side of the equality in Eqs. (2.33) and (2.34) are, in absolute magnitude, very much greater than the term on the left of these equalities [3]. Thus in the H2ßBr2 experiment it is assumed that steady-state concentrations of radicals are approached and the concentrations for H and Br are found to be (Br)  (k1 /k5 )1/ 2 (Br2 )1/ 2 (H) 

k2 (k1 /k5 )1/ 2 (H 2 )(Br2 )1/ 2 k3 (Br2 )  k4 (HBr)

(2.35) (2.36)

By substituting these values in the rate expression for the formation of HBr [Eq. (2.32)], one obtains 2 k (k /k )1/ 2 (H 2 )(Br2 )1/ 2 d (HBr)  2 1 5 dt 1  [k4 (HBr)/k3 (Br2 )] which is the exact form found experimentally [Eq. (2.31)]. Thus, kexp  2 k2 (k1 /k5 )1/ 2 ,

k exp  k4 /k3

(2.37)

57

Chemical Kinetics

Consequently, it is seen, from the measurement of the overall reaction rate and the steady-state approximation, that values of the rate constants of the intermediate radical reactions can be determined without any measurement of radical concentrations. Values kexp and kexp evolve from the experimental measurements and the form of Eq. (2.31). Since (k1/k5) is the inverse of the equilibrium constant for Br2 dissociation and this value is known from thermodynamics, k2 can be found from kexp. The value of k4 is found from k2 and the equilibrium constant that represents reactions (2.2) and (2.4), as written in the H2ßBr2 reaction scheme. From the experimental value of kexp and the calculated value of k4, the value k3 can be determined. The steady-state approximation, found to be successful in application to this straight-chain process, can be applied to many other straight-chain processes, chain reactions with low degrees of branching, and other types of non-chain systems. Because the rates of the propagating steps greatly exceed those of the initiation and termination steps in most, if not practically all, of the straight-chain systems, the approximation always works well. However, the use of the approximation in the initiation or termination phase of a chain system, during which the radical concentrations are rapidly increasing or decreasing, can lead to substantial errors.

E. PSEUDO-FIRST-ORDER REACTIONS AND THE “FALL-OFF” RANGE As mentioned earlier, practically all reactions are initiated by bimolecular collisions; however, certain bimolecular reactions exhibit first-order kinetics. Whether a reaction is first- or second-order is particularly important in combustion because of the presence of large radicals that decompose into a stable species and a smaller radical (primarily the hydrogen atom). A prominent combustion example is the decay of a paraffinic radical to an olefin and an H atom. The order of such reactions, and hence the appropriate rate constant expression, can change with the pressure. Thus, the rate expression developed from one pressure and temperature range may not be applicable to another range. This question of order was first addressed by Lindemann [4], who proposed that first-order processes occur as a result of a two-step reaction sequence in which the reacting molecule is activated by collisional processes, after which the activated species decomposes to products. Similarly, the activated molecule could be deactivated by another collision before it decomposes. If A is considered the reactant molecule and M its nonreacting collision partner, the Lindemann scheme can be represented as follows: k

f ⎯⎯⎯ ⎯⎯ → A*  M AM← ⎯

kb

k

p A* ⎯ ⎯⎯ → products

(2.38) (2.39)

58

Combustion

The rate of decay of species A is given by d (A)  kf (A)(M)  kb (A* )(M) dt

(2.40)

and the rate of change of the activated species A* is given by d (A* )  kf (A)(M)  kb (A* )(M)  kp (A* ) ≅ 0 dt

(2.41)

Applying the steady-state assumption to the activated species equation gives (A* ) 

kf (A)(M) kb (M)  kp

(2.42)

Substituting this value of (A*) into Eq. (2.40), one obtains 

kf kp (M) 1 d (A)   kdiss (A) dt kb (M)  kp

(2.43)

where kdiss is a function of the rate constants and the collision partner concentration—that is, a direct function of the total pressure if the effectiveness of all collision partners is considered the same. Owing to size, complexity, and the possibility of resonance energy exchange, the effectiveness of a collision partner (third body) can vary. Normally, collision effectiveness is not a concern, but for some reactions specific molecules may play an important role [15]. At high pressures, kb(M) kp and kdiss,∞ ≡

kf k p kb

 Kkp

(2.44)

where kdiss, becomes the high-pressure-limit rate constant and K is the equilibrium constant (kf / kb). Thus at high pressures the decomposition process becomes overall first-order. At low pressure, kb(M)

kp as the concentrations drop and kdiss,0 ≡ kf (M)

(2.45)

where kdiss,0 is the low-pressure-limit rate constant. The process is then secondorder by Eq. (2.43), simplifying to d(A)/dt  kf(M)(A). Note the presence of the concentration (A) in the manner in which Eq. (2.43) is written. Many systems fall in a region of pressures (and temperatures) between the high- and low-pressure limits. This region is called the “fall-off range,” and

59

Chemical Kinetics

its importance to combustion problems has been very adequately discussed by Troe [16]. The question, then, is how to treat rate processes in the fall-off range. Troe proposed that the fall-off range between the two limiting rate constants be represented using a dimensionless pressure scale (kdiss,0 /kdiss,∞ )  kb (M)/kp

(2.46)

in which one must realize that the units of kb and kp are different so that the right-hand side of Eq. (2.46) is dimensionless. Substituting Eq. (2.44) into Eq. (2.43), one obtains kb (M)/kp kdiss kb (M)   kdiss,∞ kb (M)  kp [kb (M)/kp ] 1

(2.47)

kdiss,0 /kdiss,∞ kdiss  kdiss,∞ 1  (kdiss,0 /kdiss,∞ )

(2.48)

or, from Eq. (2.46)

For a pressure (or concentration) in the center of the fall-off range, (kdiss,0 / kdiss,)  1 and kdiss  0.5kdiss,∞

(2.49)

Since it is possible to write the products designated in Eq. (2.39) as two species that could recombine, it is apparent that recombination reactions can exhibit pressure sensitivity; so an expression for the recombination rate constant similar to Eq. (2.48) can be developed [16]. The preceding discussion stresses the importance of properly handling rate expressions for thermal decomposition of polyatomic molecules, a condition that prevails in many hydrocarbon oxidation processes. For a detailed discussion on evaluation of low- and high-pressure rate constants, again refer to Ref. [16]. Another example in which a pseudo-first-order condition can arise in evaluating experimental data is the case in which one of the reactants (generally the oxidizer in a combustion system) is in large excess. Consider the arbitrary process AB → D

(2.50)

where (B) (A). The rate expression is d (A) d (D)   k (A)(B) dt dt

(2.51)

60

Combustion

Since (B) (A), the concentration of B does not change appreciably and k(B) would appear as a constant. Then Eq. (2.51) becomes d (A) d (D)   k(A) dt dt

(2.52)

where k  k(B). Equation (2.52) could represent experimental data because there is little dependence on variations in the concentration of the excess component B. The reaction, of course, appears overall first-order. One should keep in mind, however, that k contains a concentration and is pressure-dependent. This pseudo-first-order concept arises in many practical combustion systems that are very fuel-lean; that is, O2 is present in large excess.

F. THE PARTIAL EQUILIBRIUM ASSUMPTION As will be discussed in the following chapter, most combustion systems entail oxidation mechanisms with numerous individual reaction steps. Under certain circumstances a group of reactions will proceed rapidly and reach a quasi-equilibrium state. Concurrently, one or more reactions may proceed slowly. If the rate or rate constant of this slow reaction is to be determined and if the reaction contains a species difficult to measure, it is possible through a partial equilibrium assumption to express the unknown concentrations in terms of other measurable quantities. Thus, the partial equilibrium assumption is very much like the steadystate approximation discussed earlier. The difference is that in the steady-state approximation one is concerned with a particular species and in the partial equilibrium assumption one is concerned with particular reactions. Essentially then, partial equilibrium comes about when forward and backward rates are very large and the contribution that a particular species makes to a given slow reaction of concern can be compensated for by very small differences in the forward and backward rates of those reactions in partial equilibrium. A specific example can illustrate the use of the partial equilibrium assumption. Consider, for instance, a complex reacting hydrocarbon in an oxidizing medium. By the measurement of the CO and CO2 concentrations, one wants to obtain an estimate of the rate constant of the reaction CO  OH → CO2  H

(2.53)

d (CO2 ) d (CO)   k (CO)(OH) dt dt

(2.54)

The rate expression is

Then the question is how to estimate the rate constant k without a measurement of the OH concentration. If one assumes that equilibrium exists between

61

Chemical Kinetics

the H2ßO2 chain species, one can develop the following equilibrium reactions of formation from the complete reaction scheme: 1 2

H 2  21 O2  OH,

2 K C,f,OH 

2 (OH)eq

(H 2 )eq (O2 )eq

,

H 2  21 O2  H 2 O K C,f,H2 O 

(H 2 O)eq (H 2 )eq (O2 )1eq/ 2

(2.55)

Solving the two latter expressions for (OH)eq and eliminating (H2)eq, one obtains 2 (OH)eq  (H 2 O)1/ 2 (O2 )1/ 4 [K C,f,OH /K C,f,H2 O ]1/ 2

(2.56)

and the rate expression becomes d (CO2 ) d (CO) 2 /K C,f,H2 O ]1/2 (CO)(H 2 O)1/2 (O2 )1/4   k[K C,f,OH dt dt

(2.57)

Thus, one observes that the rate expression can be written in terms of readily measurable stable species. One must, however, exercise care in applying this assumption. Equilibria do not always exist among the H2ßO2 reactions in a hydrocarbon combustion system—indeed, there is a question if equilibrium exists during CO oxidation in a hydrocarbon system. Nevertheless, it is interesting to note the availability of experimental evidence that shows the rate of formation of CO2 to be (1/4)-order with respect to O2, (1/2)-order with respect to water, and first-order with respect to CO [17, 18]. The partial equilibrium assumption is more appropriately applied to NO formation in flames, as will be discussed in Chapter 8.

G. PRESSURE EFFECT IN FRACTIONAL CONVERSION In combustion problems, one is interested in the rate of energy conversion or utilization. Thus it is more convenient to deal with the fractional change of a particular substance rather than the absolute concentration. If (M) is used to denote the concentrations in a chemical reacting system of arbitrary order n, the rate expression is d (M)  k (M)n dt

(2.58)

Since (M) is a concentration, it may be written in terms of the total density ρ and the mole or mass fraction ε; that is, (M)  ρε

(2.59)

62

Combustion

It follows that at constant temperature ρ(d ε /dt )  k (ρε)n

(2.60)

(d ε /dt )  k ε n ρ n1

(2.61)

For a constant-temperature system, ρ  P and (d ε /dt ) ∼ P n1

(2.62)

That is, the fractional change is proportional to the pressure raised to the reaction order 1.

H. CHEMICAL KINETICS OF LARGE REACTION MECHANISMS For systems with large numbers of species and reactions, the dynamics of the reaction and the interactions between species can become quite complex. In order to analyze the reaction progress of species, various diagnostics techniques have been developed. Two of these techniques are reaction rate-ofproduction analysis and sensitivity analysis. A sensitivity analysis identifies the rate limiting or controlling reaction steps, while a rate-of-production analysis identifies the dominant reaction paths (i.e., those most responsible for forming or consuming a particular species). First, as mentioned previously, for a system of reactions, Eq. (2.1) can be rewritten as n

∑ νj,i (M j )



j 1

n

∑ ν j,i (M j ),

i  1, … , m

j 1

(2.63)

where the index i refers to reactions 1 through m of the mechanism. Following Eq. (2.3), the net reaction rate for the ith reaction can then be expressed as n

n

j 1

j 1

qi  kfi ∏ (M j )νj,i kbi ∏ (M j )ν j ,i

(2.64)

From Eq. (2.3), the rate of change of concentration of a given species j resulting from both the forward and backward reactions of the ith reaction is given by ω ji  [ν j , i  νj , i ]qi  ν ji qi

(2.65)

63

Chemical Kinetics

Given m reactions in the mechanism, the rate of change of concentration of the jth species resulting from all m reactions is given by m

ω j  ∑ ν ji qi

(2.66)

i1

For a temporally reacting system at constant temperature, the coupled species equations are then d (M j ) dt

 ω j

j  1,… , n

(2.67)

These equations are a set of nonlinear first-order ordinary differential equations that describe the evolution of the n species as a function of time starting from a set of initial conditions (M j )t 0  (M j )0

j  1, … , n

(2.68)

Because the rates of reactions can be vastly different, the timescales of change of different species concentrations can vary significantly. As a consequence, the equations are said to be stiff and require specialized numerical integration routines for their solution [19]. Solution methods that decouple the timescales of the different species (e.g., to eliminate the fast processes if only the slow rate limiting processes are of concern) have also been developed [20, 21].

1. Sensitivity Analysis The sensitivity analysis of a system of chemical reactions consist of the problem of determining the effect of uncertainties in parameters and initial conditions on the solution of a set of ordinary differential equations [22, 23]. Sensitivity analysis procedures may be classified as deterministic or stochastic in nature. The interpretation of system sensitivities in terms of first-order elementary sensitivity coefficients is called a local sensitivity analysis and typifies the deterministic approach to sensitivity analysis. Here, the first-order elementary sensitivity coefficient is defined as the gradient ∂(M j )/∂αi where (Mj) is the concentration of the jth species at time t and αi is the ith input parameter (e.g., rate constant) and the gradient is evaluated at a set of nominal parameter values α . Although, the linear sensitivity coefficients ∂(Mj)/∂αi provide direct information on the effect of a small perturbation in

64

Combustion

each parameter about its nominal value on each concentration j they do not necessarily indicate the effect of simultaneous, large variations in all parameters on each species concentration. An analysis that accounts for simultaneous parameter variations of arbitrary magnitude can be termed a global sensitivity analysis. This type of analysis produces coefficients that have a measure of sensitivity over the entire admissible range of parameter variation. Examples include the “brute force” method where a single parameter value is changed and the time history of species profiles with and without the modification are compared. Other methods are the FAST method [24], Monte Carlo methods [25], and Pattern methods [26]. The set of equations described by Eq. (2.67) can be rewritten to show the functional dependence of the right-hand side of the equation as d (M j ) dt

 f j ⎡⎢ (M j )α ⎤⎥ , ⎣ ⎦

j  1, … , n

(2.69)

where fi is the usual nonlinear first-order, second-order, or third-order function of species concentrations. The parameter vector α includes all physically definable input parameters of interest (e.g., rate constants, equilibrium constants, initial concentrations, etc.), all of which are treated as constant. For a local sensitivity analysis, Eq. (2.69) may be differentiated with respect to the parameters α to yield a set of linear coupled equations in terms of the elementary sensitivity coefficients, ∂(Mi)/∂αj. n ∂f j ∂(M s ) ∂ ⎛⎜ ∂(M j ) ⎞⎟⎟ ∂f j ⎜⎜ , ∑ ⎟⎟  ∂t ⎜⎝ ∂αi ⎠ ∂αi s1 ∂(M s ) ∂αi

i  1, … , m

(2.70)

Since the quantities ∂fi/∂(Mj) are generally required during the solution of Eq. (2.69), the sensitivity equations are conveniently solved simultaneously with the species concentration equations. The initial conditions for Eq. (2.70) result from mathematical consideration versus physical consideration as with Eq. (2.69). Here, the initial condition [∂(M j )/∂αi ]t 0 is the zero vector, unless αi is the initial concentration of the jth species, in which case the initial condition is a vector whose components are all zero except the jth component, which has a value of unity. Various techniques have been developed to solve Eq. (2.70) [22, 23]. It is often convenient, for comparative analysis, to compute normalized sensitivity coefficients ∂ ln(M j ) αi ∂(M j )  (M j ) ∂αi ∂ ln αi

(2.71)

65

Chemical Kinetics

αi

∂(M j ) ∂αi



∂(M j ) ∂ ln αi

(2.72)

and thus remove any artificial variations to the magnitudes of (Mj) or αi. Thus, the interpretation of the first-order elementary sensitivities of Eq. (2.71) is simply the percentage change in a species concentration due to a percentage change in the parameter αi at a given time t. Since it is common for species concentrations to vary over many orders of magnitude during the course of a reaction, much of the variation in the normalized coefficients of Eq. (2.71) may result from the change in the species concentration. The response of a species concentration in absolute units due to a percentage change in αi as given in Eq. (2.72) is an alternative normalization procedure. For a reversible reaction the forward and backward rate constants are related to the equilibrium constant. Thus, a summation of elementary sensitivity coefficients for the forward and backward rate constants of the same reaction is an indication of the importance of the net reaction in the mechanism, whereas the difference in the two sensitivity coefficients is an indicator of the importance of the equilibrium constant. In addition to the linear sensitivity coefficients described above, various other types of sensitivity coefficients have been studied to probe underlying relationships between input and output parameters of chemical kinetic models. These include higher-order coefficients, Green’s function coefficients, derived coefficients, feature coefficients, and principal components. Their descriptions and applications can be found in the literature [22, 23, 27, 28].

2. Rate of Production Analysis A rate-of-production analysis considers the percentage of the contributions of different reactions to the formation or consumption of a particular chemical species. The normalized production contributions of a given reaction to a particular species is given by C jip 

max(ν j , i , 0)qi m

(2.73)

∑ max(ν j,i , 0)qi i1

The normalized destruction contribution is given by C dji 

min(ν j , i , 0)qi m

∑ min(ν j,i , 0)qi i1

(2.74)

66

Combustion

The function max (x, y) implies the use of the maximum value between the two arguments x and y in the calculation. A similar definition applies to min (x, y). A local reaction flow analysis considers the formation and consumption of species locally; that is, at specific times in time-dependent problems or at specific locations in steady spatially dependent problems [29–31]. An integrated reaction flow analysis considers the overall formation or consumption during the combustion process [29, 30]. Here, the results for homogeneous time-dependent systems are integrated over the whole time, while results from steady spatially dependent systems are integrated over the reaction zone. From such results the construction of reaction flow diagrams may be developed to understand which reactions are most responsible for producing or consuming species during the reaction, that is, which are the fastest reactions among the mechanism.

3. Coupled Thermal and Chemical Reacting Systems Since combustion processes generate significant sensible energy during reaction, the species conservation equations of Eq. (2.67) become coupled to the energy conservation equation through the first law of thermodynamics. If the reaction system is treated as a closed system of fixed mass, only the species and energy equations need to be considered. Consider a system with total mass n

m  ∑ mj

(2.75)

j 1

where mj is the mass of the jth species. Overall mass conservation yields dm/dt  0, and therefore the individual species are produced or consumed as given by dm j dt

 V ω j MWj

(2.76)

where V is the volume of the system and MWj is the molecular weight of the jth species. Since the total mass is constant, Eq. (2.76) can be written in terms of the mass fractions

Yj 

mj m



mj n

∑ ms s1

(2.77)

67

Chemical Kinetics

Note that ΣYi  1. The mole fraction is defined as Xj 

Nj



N

Nj

(2.78)

n

∑ Ns s1

with ΣXi  1. Mass fractions can be related to mole fractions Yj 

X j MWj

(2.79)

MW

where n

MW  ∑ X j MWj  j 1

1 n

∑ (Y j /MWj )

(2.80)

j 1

Introducing the mass fraction into Eq. (2.76) yields dY j dt



ω j MW j ρ

,

j  1, ... , n

(2.81)

where ρ  m/V. For a multi-component gas, the mean mass density is defined by n

ρ  ∑ (M j )MWj j 1

For an adiabatic constant pressure system the first law reduces to dh  0 since h  e  Pv, dh  de  vdP  Pdv and de  Pdv where v is the specific volume (V/m). For a mixture, the total enthalpy may be written as n

h  ∑ Yj h j

(2.82)

n ⎛ dh j dY j ⎞⎟ ⎟⎟ dh  ∑ ⎜⎜⎜Y j  hj dt dt ⎟⎠ ⎝ j 1 ⎜

(2.83)

j 1

and therefore

68

Combustion

Assuming a perfect gas mixture, dh j  c p, j dT

(2.84)

and therefore n

dh  ∑ Y j c p, j j 1

n dY j dT  ∑ hj 0 dt dt j 1

(2.85)

Defining the mass weighted specific heat of the mixture as n

∑ c p , j Y j  cP

(2.86)

j 1

and substituting Eq. (2.81) into Eq. (2.85) yields the system energy equation written in terms of the temperature n

dT  dt

∑ h j ω j MW j j 1

ρc p

(2.87)

Equations (2.81) and (2.87) form a coupled set of equations, which describe the evolution of species and mixture temperature during the course of a chemical reaction. The solution procedure to this equation is the same as described for the temporal isothermal species equations described above. In addition, the associated temperature sensitivity equation can be simply obtained by taking the derivative of Eq. (2.87) with respect to each of the input parameters to the model. The governing equations for similar types of homogeneous reaction systems can be developed for constant volume systems, and stirred and plug flow reactors as described in Chapters 3 and 4 and elsewhere [31–37]. The solution to homogeneous systems described by Eq. (2.81) and Eq. (2.87) are often used to study reaction mechanisms in the absence of mass diffusion. These equations (or very similar ones) can approximate the chemical kinetics in flow reactor and shock tube experiments, which are frequently used for developing hydrocarbon combustion reaction mechanisms.

4. Mechanism Simplification As noted in the previous sections, the solution of a chemical kinetics problem in which a large detailed mechanism is used to describe the reaction requires the solution of one species conservation equation for each species of the

69

Chemical Kinetics

mechanism. For realistic fuels, the number of species could be large (several hundred or more), and consequently, the use of such mechanisms in analyzing problems with one or more spatial dimensions can be quite costly in terms of computational time. Thus, methods to simplify detailed reaction mechanisms retaining only the essential features have been understudy. Simplified mechanisms can also provide additional insight into the understanding of the chemistry by decreasing the complexities of a large detailed mechanism. Steady-state and partial equilibrium assumptions have been used to generate reduced mechanisms [38, 39] and sensitivity analysis techniques have been used to generate skeletal mechanisms [23]. An eigenvalue analysis of the Jacobian associated with the differential equations of the system reveals information about the timescales of the chemical reaction and about species in steady-state or reactions in partial equilibrium [24]. The eigenvalues can be used to separate the species with fast and slow timescales, and thus, the system may be simplified, for example, by eliminating the fast species by representing them as functions of the slow ones. Examples of such approaches to mechanism simplification are readily available and the reader is referred to the literature for more details [40–45].

PROBLEMS (Those with an asterisk require a numerical solution and use of an appropriate software program—see Appendix I.) 1. For a temperature of 1000 K, calculate the pre-exponential factor in the specific reaction rate constant for (a) any simple bimolecular reaction and (b) any simple unimolecular decomposition reaction following transition state theory. 2. The decomposition of acetaldehyde is found to be overall first-order with respect to the acetaldehyde and to have an overall activation energy of 60 kcal/mol. Assume the following hypothetical sequence to be the chain decomposition mechanism of acetaldehyde: i

k

i

1 (1) CH3 CHO ⎯ ⎯⎯ → 0.5CH3CO  0.5CH3  0.5CO  0.5H 2

i

i

k

(2) CH3 CO ⎯ ⎯2 ⎯ → CH3  CO i

k

i

3 (3) CH3  CH3 CHO ⎯ ⎯⎯ → CH 4  CH 3 CO

i

i

k

(4) CH3  CH3 CO ⎯ ⎯4 ⎯ → minor products For these conditions, (a) List the type of chain reaction and the molecularity of each of the four reactions. (b) Show that these reaction steps would predict an overall reaction order of 1 with respect to the acetaldehyde.

70

Combustion

(c) Estimate the activation energy of reaction (2), if El  80, E3  10, and E4  5 kcal/mol. Hint: El is much larger than E2, E3, and E4. 3. Assume that the steady state of (Br) is formally equivalent to partial equilibrium for the bromine radical chain-initiating step and recalculate the form of Eq. (2.37) on this basis. 4. Many early investigators interested in determining the rate of decomposition of ozone performed their experiments in mixtures of ozone and oxygen. Their observations led them to write the following rate expression: d (O3 )/dt  kexp [(O3 )2 /(O2 )] The overall thermodynamic equation for ozone conversion to oxygen is 2O3 → 3O2 The inhibiting effect of the oxygen led many to expect that the decomposition followed the chain mechanism k

1 M  O3 ⎯ ⎯⎯ → O2  O  M

k

2 M  O  O2 ⎯ ⎯⎯ → O3  M

k

3 O  O3 ⎯ ⎯⎯ → 2O 2

(a) If the chain mechanisms postulated were correct and if k2 and k3 were nearly equal, would the initial mixture concentration of oxygen have been much less than or much greater than that of ozone? (b) What is the effective overall order of the experimental result under these conditions? (c) Given that kexp was determined as a function of temperature, which of the three elementary rate constants is determined? Why? (d) What type of additional experiment should be performed in order to determine all the elementary rate constants? 5. A strong normal shock wave is generated in a shock tube filled with dry air at standard temperature and pressure (STP). The oxygen and nitrogen behind the shock wave tend to react to form nitric oxide. Calculate the mole fraction of nitric oxide that ultimately will form, assuming that the elevated temperature and pressure created by the shock are sustained indefinitely. Calculate the time in milliseconds after the passage of the shock for the attainment of 50% of the ultimate amount; this time may be termed the “chemical relaxation time” for the shock process. Calculate the corresponding “relaxation distance,” that is, the distance from the shock wave where 50% of the ultimate chemical change has occurred.

71

Chemical Kinetics

Use such reasonable approximations as: (1) Air consists solely of nitrogen and oxygen in exactly 4:1 volume ratio; (2) other chemical “surface” reactions can be neglected because of the short times; (4) ideal shock wave relations for pure air with constant specific heats may be used despite the formation of nitric oxide and the occurrence of high temperature. Do the problem for two shock strengths, M  6 and M  7. The following data may be used: i. At temperatures above 1250 K, the decomposition of pure nitric oxide is a homogeneous second order reaction: ⎛ mol ⎞1 k  2.2  1014 exp(78,200/RT ) ⎜⎜ 3 ⎟⎟⎟ s1 ⎜⎝ cm ⎠ See: Wise, H. and Frech, M. Fr, Journal of Chemical Physics, 20, 22 and 1724 (1952). ii. The equilibrium constant for nitric oxide in equilibrium with nitrogen and oxygen is tabulated as follows:

T°

Kp

1500

0.00323

1750

0.00912

2000

0.0198

2250

0.0364

2500

0.0590

2750

0.0876

See: Gaydon, A. G. and Wolfhard, H. G., “Flames: Their Structure, Radiation and Temperature,” Chapman and Hall, 1970, page 274. 6. Gaseous hydrazine decomposes in a flowing system releasing energy. The decomposition process follows first order kinetics. The rate of change of the energy release is of concern. Will this rate increase, decrease, or remain the same with an increase in pressure? 7. Consider the hypothetical reaction AB → CD The reaction as shown is exothermic. Which has the larger activation energy, the exothermic forward reaction or its backward analog? Explain.

72

Combustion

8. The activation energy for dissociation of gaseous HI to H2 and I2 (gas) is 185.5 kJ/mol. If the ΔH of ,298 for HI is 5.65 kJ/mol, what is the Ea for the reaction H2  I2 (gas) → HI (gas). 9. From the data in Appendix C, determine the rate constant at 1000 K for the reaction. k

f H 2  OH ⎯ ⎯⎯ → H2 O  H

Then, determine the rate constant of the reverse reaction. 10. Consider the chemical reaction of Problem 9. It is desired to find an expression for the rate of formation of the water vapor when all the radicals can be considered to be in partial equilibrium. 11. Consider the first order decomposition of a substance A to products. At constant temperature, what is the half-life of the substance? 12.*A proposed mechanism for the reaction between H2 and Cl2 to form HCl is given below. Cl2  M H2  M H  Cl2 Cl  H 2 H  Cl  M

    

Cl  Cl  M HHM HCl  Cl HCl  H HCl  M

Calculate and plot the time-dependent species profiles for an initial mixture of 50% H2 and 50% Cl2 reacting at a constant temperature and pressure of 800 K and 1 atm, respectively. Consider a reaction time of 200 ms. Perform a sensitivity analysis and plot the sensitivity coefficients of the HCl concentration with respect to each of the rate constants. Rank-order the importance of each reaction on the HCl concentration. Is the H atom concentration in steady-state? 13.*High temperature NO formation in air results from a thermal mechanism (see Chapter 8) described by the two reactions. N 2  O  NO  N N  O2  NO  O Add to this mechanism the reaction for O2 dissociation O2  M  O  O  M and calculate the time history of NO formation at a constant temperature and pressure of 2500 K and 1 atm, respectively. Develop a mechanism that has separate reactions for the forward and backward directions. Obtain one

Chemical Kinetics

73

of the rate constants for each reaction from Appendix C and evaluate the other using the thermodynamic data of Appendix A. Plot the species profiles of NO and O as a function of time, as well as the sensitivity coefficients of NO with respect to each of the mechanism rate constants. What is the approximate time required to achieve the NO equilibrium concentration? How does this time compare to residence times in flames or in combustion chambers?

REFERENCES 1. Benson, S. W., “The Foundations of Chemical Kinetics.” McGraw-Hill, New York, 1960; Weston, R. E., and Schwartz, H. A., “Chemical Kinetics.” Prentice-Hall, Englewood Cliffs, NJ, 1972; Smith, I. W. M., “Kinetics and Dynamics of Elementary Reactions.” Butterworth, London, 1980. 1a. Laidler, K. J., “Theories of Chemical Reaction Rates.” McGraw-Hill, New York, 1969. 2. Penner, S. S., “Introduction to the Study of the Chemistry of Flow Processes,” Chap. 1. Butterworth, London, 1955. 3. Williams, F. A., “Combustion Theory,” 2nd Ed. Benjamin-Cummings, Menlo Park, California, 1985, Appendix B. 4. Lindemann, F. A., Trans. Faraday Soc. 17, 598 (1922). 5. Arrhenius, S., Phys. Chem. 4, 226 (1889). 6. Vincenti, W. G., and Kruger, C. H. Jr., “Introduction to Physical Gas Dynamics,” Chap. 7. Wiley, New York, 1965. 7. Dryer, F. L., Naegeli, D. W., and Glassman, I., Combust. Flame 17, 270 (1971). 8. Fontijn, A., and Zellner, R., in “Reactions of Small Transient Species.” (A. Fontijn and M. A. A. Clyne, eds.). Academic Press, Orlando, Florida, 1983. 9. Kaufman, F., Proc. Combust. Inst. 15, 752 (1974). 10. Golden, D. M., in “Fossil Fuel Combustion.” (W. Bartok and A. F. Sarofirn, eds.), p. 49. Wiley (Interscience), New York, 1991. 11. Landau, L., and Teller, E., Phys. Z. Sowjet. 10(l), 34 (1936). 12. Bodenstein, M., Phys. Chem. 85, 329 (1913). 13. Dainton, F. S., “Chain Reactions: An Introduction,” Methuen, London, 1956. 14. Bradley, J. N., “Flame and Combustion Phenomena,” Chap. 2. Methuen, London, 1969. 15. Yetter, R. A., Dryer, F. L., and Rabitz, H., 7th Int. Symp. Gas Kinet., P. 231. Gottingen, 1982. 16. Troe, J., Proc. Combust. Inst. 15, 667 (1974). 17. Dryer, F. L., and Glassman, I., Proc. Combust. Inst. 14, 987 (1972). 18. Williams, G. C., Hottel, H. C., and Morgan, A. G., Proc. Combust. Instit. 12, 913 (1968). 19. Hindmarsh, A. C., ACM Sigum Newsletter 15, 4 (1980). 20. Lam, S. H., in “Recent Advances in the Aerospace Sciences.” (Corrado. Casci, ed.), pp. 3–20. Plenum Press, New York and London, 1985. 21. Lam, S. H., and Goussis, D. A., Proc. Combust. Inst. 22, 931 (1989). 22. Rabitz, H., Kramer, M., and Dacol, D., Ann. Rev. Phys. Chem. 34, 419 (1983). 23. Turányi, T., J. Math. Chem. 5, 203 (1990). 24. Cukier, R. I., Fortuin, C. M., Shuler, K. E., Petschek, A. G., and Schaibly, J. H., J. Chem. Phys. 59, 3873 (1973). 25. Stolarski, R. S., Butler, D. M., and Rundel, R. D., J. Geophys. Res. 83, 3074 (1978). 26. Dodge, M. C., and Hecht, T. A., Int. J. Chem. Kinet. 1, 155 (1975). 27. Yetter, R. A., Dryer, F. L., and Rabitz, H., Combust. Flame 59, 107 (1985). 28. Turányi, T., Reliab. Eng. Syst. Saf. 57, 4 (1997). 29. Warnatz, J., Proc. Combust. Inst. 18, 369 (1981).

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Combustion

30. Warnatz, J., Mass, U., and Dibble, R. W., “Combustion.” Springer-Verlag, Berlin, 1996. 31. Kee, R. J. Rupley, F. M. Miller, J. A. Coltrin, M. E. Grcar, J. F. Meeks, E. Moffat, H. K. Lutz, A. E. Dixon-Lewis, G. Smooke, M. D. Warnatz, J. Evans, G. H. Larson, R. S. Mitchell, R. E. Petzold, L. R. Reynolds, W. C. Caracotsios, M. Stewart, W. E. Glarborg, P. Wang, C. McLellan, C. L. Adigun, O. Houf, W. G. Chou, C. P. Miller, S. F. Ho, P. Young, P. D. and Young D. J., “CHEMKIN Release 4.0.2, Reaction Design”, San Diego, CA, 2005. 32. Kee, R. J. Miller, J. A. and Jefferson, T. H. “Chemkin: A General-Purpose, ProblemIndependent Transportable, Fortran Chemical Kinetics Code Package”, Sandia National Laboratories Report SAND80-8003, 1980. 33. Kee, R. J. Rupley, F. M. and Miller, J. A. “Chemkin-II: A Fortran Chemical Kinetics Package for the Analysis of Gas-Phase Chemical Kinetics”, Sandia National Laboratories Report SAND89-8009, 1990. 34. Kee, R. J. Rupley, F. M. Meeks, E. and Miller, J. A. “Chemkin-III: A Fortran Chemical Kinetics Package for the Analysis of Gas-Phase Chemical and Plasma Kinetics”, Sandia National Laboratories Report SAND96-8216, 1996. 35. Lutz, E. Kee, R. J. and Miller, J. A. “SENKIN: A Fortran Program for Predicting Homogeneous Gas Phase Chemical Kinetics with Sensitivity Analysis”, Sandia National Laboratories Report 87-8248, 1988. 36. Glarborg, P., Kee, R.J., Grcar, J.F., and Miller, J.A., PSR: A Fortran Program for Modeling Well-Stirred Reactors, Sandia National Laboratories Report 86-8209, 1986. 37. Turns, S. R., “An Introduction to Combustion,” 2nd Ed. McGraw Hill, Boston, 2000. 38. Smooke, M. D., Reduced Kinetic Mechanisms and Asymptotic Approximations for Methane– Air Flames “Lecture Notes in Physics 384”. Springer-Verlag, New York, 1991. 39. Varatharajan, B., and Williams, F. A., J. Propul. Power 18, 352 (2002). 40. Lam, S. H., Combust. Sci. Technol. 89, 375 (1993). 41. Mass, U., and Pope, S. B., Combust. Flame 88, 239 (1992). 42. Mass, U., and Pope, S. B., Proc. Combust. Inst. 24, 103 (1993). 43. Li, G., and Rabitz, H., Chem. Eng. Sci. 52, 4317 (1997). 44. Büki, A., Perger, T., Turányi, T., and Maas, U., J. Math. Chem. 31, 345 (2002). 45. Lu, T., and Law, C. K., Proc. Combust. Inst. 30, 1333 (2005).

Chapter 3

Explosive and General Oxidative Characteristics of Fuels A. INTRODUCTION In the previous chapters, the fundamental areas of thermodynamics and chemical kinetics were reviewed. These areas provide the background for the study of very fast reacting systems, termed explosions. In order for flames (deflagrations) or detonations to propagate, the reaction kinetics must be fast—that is, the mixture must be explosive.

B. CHAIN BRANCHING REACTIONS AND CRITERIA FOR EXPLOSION Consider a mixture of hydrogen and oxygen stored in a vessel in stoichiometric proportions and at a total pressure of 1 atm. The vessel is immersed in a thermal bath kept at 500°C (773 K), as shown in Fig. 3.1. If the vessel shown in Fig. 3.1 is evacuated to a few millimeters of mercury (torr) pressure, an explosion will occur. Similarly, if the system is pressurized to 2 atm, there is also an explosion. These facts suggest explosive limits. If H2 and O2 react explosively, it is possible that such processes could occur in a flame, which indeed they do. A fundamental question then is: What governs the conditions that give explosive mixtures? In order to answer this question,

Supply

Pump

H2, O2 1 atm 500C FIGURE 3.1 Experimental configuration for the determination of H2¶O2 explosion limits.

75

76

Combustion

it is useful to reconsider the chain reaction as it occurs in the H2 and Br2 reaction: H 2  Br2 → 2HBr

(the overall reaction)

M  Br2 → 2Br  M

(chain initiating step)

Br  H 2 → HBr  H ⎪⎫⎪ ⎪ H  Br2 → HBr  Br ⎬ ⎪ H  HBr → H 2  Br ⎪⎪⎪⎭

(chain propagating steps)

M  2Br → Br2  M

(chain terminating step)

There are two means by which the reaction can be initiated—thermally or photochemically. If the H2¶Br2 mixture is at room temperature, a photochemical experiment can be performed by using light of short wavelength; that is, high enough hν to rupture the Br¶Br bond through a transition to a higher electronic state. In an actual experiment, one makes the light source as weak as possible and measures the actual energy. Then one can estimate the number of bonds broken and measure the number of HBr molecules formed. The ratio of HBr molecules formed per Br atom created is called the photoyield. In the room-temperature experiment one finds that (HBr)/(Br) ∼ 0.01

1 and, of course, no explosive characteristic is observed. No explosive characteristic is found in the photolysis experiment at room temperature because the reaction Br  H 2 → HBr  H is quite endothermic and therefore slow. Since the reaction is slow, the chain effect is overtaken by the recombination reaction M  2Br → Br2  M Thus, one sees that competitive reactions appear to determine the overall character of this reacting system and that a chain reaction can occur without an explosion. For the H2¶Cl2 system, the photoyield is of the order 104 to 107. In this case the chain step is much faster because the reaction Cl  H 2 → HCl  H

Explosive and General Oxidative Characteristics of Fuels

77

has an activation energy of only 25 kJ/mol compared to 75 kJ/mol for the corresponding bromine reaction. The fact that in the iodine reaction the corresponding step has an activation energy of 135 kJ/mol gives credence to the notion that the iodine reaction does not proceed through a chain mechanism, whether it is initiated thermally or photolytically. It is obvious, then, that only the H2¶Cl2 reaction can be exploded photochemically, that is, at low temperatures. The H2¶Br2 and H2¶I2 systems can support only thermal (high-temperature) explosions. A thermal explosion occurs when a chemical system undergoes an exothermic reaction during which insufficient heat is removed from the system so that the reaction process becomes selfheating. Since the rate of reaction, and hence the rate of heat release, increases exponentially with temperature, the reaction rapidly runs away; that is, the system explodes. This phenomenon is the same as that involved in ignition processes and is treated in detail in the chapter on thermal ignition (Chapter 7). Recall that in the discussion of kinetic processes it was emphasized that the H2¶O2 reaction contains an important, characteristic chain branching step, namely, H  O2 → OH  O which leads to a further chain branching system O  H 2 → OH  H OH  H 2 → H 2 O  H The first two of these three steps are branching, in that two radicals are formed for each one consumed. Since all three steps are necessary in the chain system, the multiplication factor, usually designated α, is seen to be greater than 1 but less than 2. The first of these three reactions is strongly endothermic; thus it will not proceed rapidly at low temperatures. So, at low temperatures an H atom can survive many collisions and can find its way to a surface to be destroyed. This result explains why there is steady reaction in some H2¶O2 systems where H radicals are introduced. Explosions occur only at the higher temperatures, where the first step proceeds more rapidly. It is interesting to consider the effect of the multiplication as it may apply in a practical problem such as that associated with automotive knock. However extensive the reacting mechanism in a system, most of the reactions will be bimolecular. The pre-exponential term in the rate constant for such reactions has been found to depend on the molecular radii and temperature, and will generally be between 4  1013 and 4  1014 cm3mol1s1. This appropriate assumption provides a ready means for calculating a collision frequency. If the state quantities in the knock regime lie in the vicinity of 1200 K and 20 atm and if nitrogen is assumed to be the major component in the gas mixture, the density

78

Combustion

of this mixture is of the order of 6 kg/m3 or approximately 200 mol/m3. Taking the rate constant pre-exponential as 1014 cm3mol1s1 or 108 m3mol1s1, an estimate of the collision frequency between molecules in the mixture is (108 m 3 mol1s1 )(200 mol/m 3 )  2  1010 collisions/s For arithmetic convenience, 1010 will be assumed to be the collision frequency in a chemical reacting system such as the knock mixture loosely defined. Now consider that a particular straight-chain propagating reaction ensues, that the initial chain particle concentration is simply 1, and that 1 mol or 1019 molecules/cm3 exist in the system. Thus all the molecules will be consumed in a straight-chain propagation mechanism in a time given by 1019 molecules cm 3 1  10  109 s 3 1 molecule cm 10 collisions s or approximately 30 years, a preposterous result. Specifying α as the chain branching factor, then, the previous example was for the condition α  1. If, however, pure chain branching occurs under exactly the same conditions, then α  2 and every radical initiating the chain system creates two, which create four, and so on. Then 1019 molecules/cm3 are consumed in the following number of generations (N): 2 N  1019 or N  63 Thus the time to consume all the particles is 63 1  10  63  1010 s 1 10 or roughly 6 ns. If the system is one of both chain branching and propagating steps, α could equal 1.01, which would indicate that one out of a hundred reactions in the system is chain branching. Moreover, hidden in this assumption is the effect of the ordinary activation energy in that not all collisions cause reaction. Nevertheless, this point does not invalidate the effect of a small amount

Explosive and General Oxidative Characteristics of Fuels

79

of chain branching. Then, if α  1.01, the number of generations N to consume the mole of reactants is 1.01N  1019 N ≅ 4400 Thus the time for consumption is 44  108 s or approximately half a microsecond. For α  1.001, or one chain branching step in a thousand, N  43,770 and the time for consumption is approximately 4 ms. From this analysis one concludes that if one radical is formed at a temperature in a prevailing system that could undergo branching and if this branching system includes at least one chain branching step and if no chain terminating steps prevent run away, then the system is prone to run away; that is, the system is likely to be explosive. To illustrate the conditions under which a system that includes chain propagating, chain branching, and chain terminating steps can generate an explosion, one chooses a simplified generalized kinetic model. The assumption is made that for the state condition just prior to explosion, the kinetic steady-state assumption with respect to the radical concentration is satisfactory. The generalized mechanism is written as follows: k

1 M ⎯ ⎯⎯ →R

(3.1)

2 R  M ⎯ ⎯⎯ → αR  M′

(3.2)

k

k

3 R  M ⎯ ⎯⎯ →PR

k

4 R  M ⎯ ⎯⎯ →I

k

5 R  O2  M ⎯ ⎯⎯ → RO2  M

k

6 R ⎯ ⎯⎯ → I

(3.3) (3.4) (3.5) (3.6)

Reaction (3.1) is the initiation step, where M is a reactant molecule forming a radical R. Reaction (3.2) is a particular representation of a collection of propagation steps and chain branching to the extent that the overall chain branching ratio can be represented as α. M is another reactant molecule and α has any value greater than 1. Reaction (3.3) is a particular chain propagating step forming a product P. It will be shown in later discussions of the hydrocarbon–air

80

Combustion

reacting system that this step is similar, for example, to the following important exothermic steps in hydrocarbon oxidation: H 2  OH → H 2 O  H ⎪⎫⎪ ⎬ CO  OH → CO2  H ⎪⎪⎭

(3.3a)

Since a radical is consumed and formed in reaction (3.3) and since R represents any radical chain carrier, it is written on both sides of this reaction step. Reaction (3.4) is a gas-phase termination step forming an intermediate stable molecule I, which can react further, much as M does. Reaction (3.5), which is not considered particularly important, is essentially a chain terminating step at high pressures. In step (5), R is generally an H radical and RO2 is HO2, a radical much less effective in reacting with stable (reactant) molecules. Thus reaction (3.5) is considered to be a third-order chain termination step. Reaction (3.6) is a surface termination step that forms minor intermediates (I) not crucial to the system. For example, tetraethyllead forms lead oxide particles during automotive combustion; if these particles act as a surface sink for radicals, reaction (3.6) would represent the effect of tetraethyllead. The automotive cylinder wall would produce an effect similar to that of tetraethyllead. The question to be considered is what value of α is necessary for the system to be explosive. This explosive condition is determined by the rate of formation of a major product, and P (products) from reaction (3.3) is the obvious selection for purposes here. Thus d (P)  k3 (R)(M) dt

(3.3b)

The steady-state assumption discussed in the consideration of the H2¶Br2 chain system is applied for determination of the chain carrier concentration (R): d (R)  k1 (M)  k2 (α  1)(R)(M)  k4 (R)(M) dt  k5 (O2 ) (R) (M)  k6 (R)  0

(3.7)

Thus, the steady-state concentration of (R) is found to be (R) 

k1 (M) k4 (M)  k5 (O2 )(M)  k6  k2 (α  1)(M)

(3.8)

Substituting Eq. (3.8) into Eq. (3.3b), one obtains k1k3 (M)2 d (P)  dt k4 (M)  k5 (O2 )(M)  k6  k2 (α  1)(M)

(3.9)

Explosive and General Oxidative Characteristics of Fuels

81

The rate of formation of the product P can be considered to be infinite—that is, the system explodes—when the denominator of Eq. (3.9) equals zero. It is as if the radical concentration is at a point where it can race to infinity. Note that k1, the reaction rate constant for the initiation step, determines the rate of formation of P, but does not affect the condition of explosion. The condition under which the denominator would become negative implies that the steady-state approximation is not valid. The rate constant k3, although regulating the major product-forming and energy-producing step, affects neither the explosion-producing step nor the explosion criterion. Solving for α when the denominator of Eq. (3.9) is zero gives the critical value for explosion; namely, αcrit  1 

k4 (M)  k5 (O2 )(M)  k6 k2 (M)

(3.10)

Assuming there are no particles or surfaces to cause heterogeneous termination steps, then αcrit = 1 +

k4 (M) + k5 (O2 )(M) k + k5 (O2 ) = 1+ 4 k2 (M) k2

(3.11)

Thus for a temperature and pressure condition where αreact αcrit, the system becomes explosive; for the reverse situation, the termination steps dominate and the products form by slow reaction. Whether or not either Eq. (3.10) or Eq. (3.11) is applicable to the automotive knock problem may be open to question, but the results appear to predict qualitatively some trends observed with respect to automotive knock. αreact can be regarded as the actual chain branching factor for a system under consideration, and it may also be the appropriate branching factor for the temperature and pressure in the end gas in an automotive system operating near the knock condition. Under the concept just developed, the radical pool in the reacting combustion gases increases rapidly when αreact αcrit, so the steady-state assumption no longer holds and Eq. (3.9) has no physical significance. Nevertheless, the steadystate results of Eq. (3.10) or Eq. (3.11) essentially define the critical temperature and pressure condition for which the presence of radicals will drive a chain reacting system with one or more chain branching steps to explosion, provided there are not sufficient chain termination steps. Note, however, that the steps in the denominator of Eq. (3.9) have various temperature and pressure dependences. It is worth pointing out that the generalized reaction scheme put forth cannot achieve an explosive condition, even if there is chain branching, if the reacting radical for the chain branching step is not regenerated in the propagating steps and this radical’s only source is the initiation step. Even though k2 is a hypothetical rate constant for many reaction chain systems within the overall network of reactions in the reacting media and hence

82

Combustion

cannot be evaluated to obtain a result from Eq. (3.10), it is still possible to extract some qualitative trends, perhaps even with respect to automotive knock. Most importantly, Eq. (3.9) establishes that a chemical explosion is possible only when there is chain branching. Earlier developments show that with small amounts of chain branching, reaction times are extremely small. What determines whether the system will explode or not is whether chain termination is faster or slower than chain branching. The value of αcrit in Eq. (3.11) is somewhat pressure-dependent through the oxygen concentration. Thus it seems that as the pressure rises, αcrit would increase and the system would be less prone to explode (knock). However, as the pressure increases, the temperature also rises. Moreover, k4, the rate constant for a bond forming step, and k5, a rate constant for a three-body recombination step, can be expected to decrease slightly with increasing temperature. The overall rate constant k2, which includes branching and propagating steps, to a first approximation, increases exponentially with temperature. Thus, as the cylinder pressure in an automotive engine rises, the temperature rises, resulting in an αcrit that makes the system more prone to explode (knock). The αcrit from Eq. (3.10) could apply to a system that has a large surface area. Tetraethyllead forms small lead oxide particles with a very large surface area, so the rate constant k6 would be very large. A large k6 leads to a large value of αcrit and hence a system unlikely to explode. This analysis supports the argument that tetraethylleads suppress knock by providing a heterogeneous chain terminating vehicle. It is also interesting to note that, if the general mechanism [Eqs. (3.1)–(3.6)] were a propagating system with α  1, the rate of change in product concentration (P) would be [d (P)/dt ]  [k1k3 (M)2 ]/[k4 (M)  k5 (O2 )(M)  k6 ] Thus, the condition for fast reaction is {k1k3 (M)2 /[k4 (M)  k5 (O2 )(M)  k6 ]} 1 and an explosion is obtained at high pressure and/or high temperature (where the rates of propagation reactions exceed the rates of termination reactions). In the photochemical experiments described earlier, the explosive condition would not depend on k1, but on the initial perturbed concentration of radicals. Most systems of interest in combustion include numerous chain steps. Thus it is important to introduce the concept of a chain length, which is defined as the average number of product molecules formed in a chain cycle or the product reaction rate divided by the system initiation rate [1]. For the previous

83

Explosive and General Oxidative Characteristics of Fuels

scheme, then, the chain length (cl) is equal to Eq. (3.9) divided by the rate expression k1 for reaction (3.1); that is, cl 

k3 (M) k4 (M)  k5 (O2)(M)  k6  k2 (α  1)(M)

(3.12)

and if there is no heterogeneous termination step, cl 

k3 k4  k5 (O2 )  k2 (α  1)

(3.12a)

If the system contains only propagating steps, α  1, so the chain length is cl 

k3 (M) k4 (M)  k5 (O2 )(M)  k6

(3.13)

and again, if there is no heterogeneous termination, cl 

k3 k4  k5 (O2 )

(3.13a)

Considering that for a steady system, the termination and initiation steps must be in balance, the definition of chain length could also be defined as the rate of product formation divided by the rate of termination. Such a chain length expression would not necessarily hold for the arbitrary system of reactions (3.1)–(3.6), but would hold for such systems as that written for the H2¶Br2 reaction. When chains are long, the types of products formed are determined by the propagating reactions alone, and one can ignore the initiation and termination steps.

C. EXPLOSION LIMITS AND OXIDATION CHARACTERISTICS OF HYDROGEN Many of the early contributions to the understanding of hydrogen–oxygen oxidation mechanisms developed from the study of explosion limits. Many extensive treatises were written on the subject of the hydrogen–oxygen reaction and, in particular, much attention was given to the effect of walls on radical destruction (a chain termination step) [2]. Such effects are not important in the combustion processes of most interest here; however, Appendix C details a complex modern mechanism based on earlier thorough reviews [3, 4].

84

Combustion

Flames of hydrogen in air or oxygen exhibit little or no visible radiation, what radiation one normally observes being due to trace impurities. Considerable amounts of OH can be detected, however, in the ultraviolet region of the spectrum. In stoichiometric flames, the maximum temperature reached in air is about 2400 K and in oxygen about 3100 K. The burned gas composition in air shows about 95–97% conversion to water, the radicals H, O, and OH comprising about one-quarter of the remainder [5]. In static systems practically no reactions occur below 675 K, and above 850 K explosion occurs spontaneously in the moderate pressure ranges. At very high pressures the explosion condition is moderated owing to a third-order chain terminating reaction, reaction (3.5), as will be explained in the following paragraphs. It is now important to stress the following points in order to eliminate possible confusion with previously held concepts and certain subjects to be discussed later. The explosive limits are not flammability limits. Explosion limits are the pressure–temperature boundaries for a specific fuel–oxidizer mixture ratio that separate the regions of slow and fast reaction. For a given temperature and pressure, flammability limits specify the lean and rich fuel–oxidizer mixture ratio beyond which no flame will propagate. Next, recall that one must have fast reactions for a flame to propagate. A stoichiometric mixture of H2 and O2 at standard conditions will support a flame because an ignition source initially brings a local mixture into the explosive regime, whereupon the established flame by diffusion heats fresh mixture to temperatures high enough to be explosive. Thus, in the early stages of any flame, the fuel–air mixture may follow a low-temperature steady reaction system and in the later stages, an explosive reaction system. This point is significant, especially in hydrocarbon combustion, because it is in the low-temperature regime that particular pollutant-causing compounds are formed. Figure 3.2 depicts the explosion limits of a stoichiometric mixture of hydrogen and oxygen. Explosion limits can be found for many different mixture ratios. The point X on Fig. 3.2 marks the conditions (773 K; 1 atm) described at the very beginning of this chapter in Fig. 3.1. It now becomes obvious that either increasing or decreasing the pressure at constant temperature can cause an explosion. Certain general characteristics of this curve can be stated. First, the third limit portion of the curve is as one would expect from simple density considerations. Next, the first, or lower, limit reflects the wall effect and its role in chain destruction. For example, HO2 radicals combine on surfaces to form H2O and O2. Note the expression developed for αcrit [Eq. (3.9)] applies to the lower limit only when the wall effect is considered as a first-order reaction of k6 → destruction was written. Although the chain destruction, since R ⎯ ⎯⎯⎯ wall features of the movement of the boundaries are not explained fully, the general shape of the three limits can be explained by reasonable hypotheses of mechanisms. The manner in which the reaction is initiated to give the boundary designated by the curve in Fig. 3.2 suggests, as was implied earlier, that the

85

Explosive and General Oxidative Characteristics of Fuels

10,000

Th

ird

lim

it

Pressure (mmHg)

1000 Fig. 3.1 Condition No explosion 100

10 Explosion

First limit 0 400

440

480

520

560

Temperature (C) FIGURE 3.2 Explosion limits of a stoichoimetric H2¶O2 mixture (after Ref. [2]).

explosion is in itself a branched chain phenomenon. Thus, one must consider possible branched chain mechanisms to explain the limits. Basically, only thermal, not photolytic, mechanisms are considered. The dissociation energy of hydrogen is less than that of oxygen, so the initiation can be related to hydrogen dissociation. Only a few radicals are required to initiate the explosion in the region of temperature of interest, that is, about 675 K. If hydrogen dissociation is the chain’s initiating step, it proceeds by the reaction H 2  M → 2H  M

(3.14)

which requires about 435 kJ/mol. The early modeling literature suggested the initiation step M  H 2  O2 → H 2 O2  M ↓ 2OH

(3.15)

because this reaction requires only 210 kJ/mol, but this trimolecular reaction has been evaluated to have only a very slow rate [6]. Because in modeling it accurately reproduces experimental ignition delay measurements under shock tube and detonation conditions [7], the most probable initiation step, except at the very highest temperature at which reaction (3.14) would prevail, could be H 2  O2 → HO2  H

(3.16)

86

Combustion

where HO2 is the relatively stable hydroperoxy radical that has been identified by mass spectroscopic analysis. There are new data that support this initiation reaction in the temperature range 1662–2097 K [7a]. The essential feature of the initiation step is to provide a radical for the chain system and, as discussed in the previous section, the actual initiation step is not important in determining the explosive condition, nor is it important in determining the products formed. Either reaction (3.14) or (3.16) provides an H radical that develops a radical pool of OH, O, and H by the chain reactions H  O2 → O  OH

(3.17)

O  H 2 → H  OH

(3.18)

H 2  OH → H 2 O  H

(3.19)

O  H 2 O → OH  OH

(3.20)

Reaction (3.17) is chain branching and 66 kJ/mol endothermic. Reaction (3.18) is also chain branching and 8 kJ/mol exothermic. Note that the H radical is regenerated in the chain system and there is no chemical mechanism barrier to prevent the system from becoming explosive. Since radicals react rapidly, their concentration levels in many systems are very small; consequently, the reverse of reactions (3.17), (3.18), and (3.20) can be neglected. Normally, reactions between radicals are not considered, except in termination steps late in the reaction when the concentrations are high and only stable product species exist. Thus, the reverse reactions (3.17), (3.18), and (3.20) are not important for the determination of the second limit [i.e., (M)  2k17/k21]; nor are they important for the steady-slow H2¶O2 and CO¶H2O¶O2 reactions. However, they are generally important in all explosive H2¶O2 and CO¶H2O¶O2 reactions. The importance of these radical– radical reactions in these cases is verified by the existence of superequilibrium radical concentrations and the validity of the partial equilibrium assumption. The sequence [Eqs. (17)–(20)] is of great importance in the oxidation reaction mechanisms of any hydrocarbon in that it provides the essential chain branching and propagating steps as well as the radical pool for fast reaction. The important chain termination steps in the static explosion experiments (Fig. 3.1) are H → wall destruction OH → wall destruction Either or both of these steps explain the lower limit of explosion, since it is apparent that wall collisions become much more predominant at lower pressure

Explosive and General Oxidative Characteristics of Fuels

87

than molecular collisions. The fact that the limit is found experimentally to be a function of the containing vessel diameter is further evidence of this type of wall destruction step. The second explosion limit must be explained by gas-phase production and destruction of radicals. This limit is found to be independent of vessel diameter. For it to exist, the most effective chain branching reaction (3.17) must be overridden by another reaction step. When a system at a fixed temperature moves from a lower to higher pressure, the system goes from an explosive to a steady reaction condition, so the reaction step that overrides the chain branching step must be more pressure-sensitive. This reasoning leads one to propose a third-order reaction in which the species involved are in large concentration [2]. The accepted reaction that satisfies these prerequisites is H  O2  M → HO2  M

(3.21)

where M is the usual third body that takes away the energy necessary to stabilize the combination of H and O2. At higher pressures it is certainly possible to obtain proportionally more of this trimolecular reaction than the binary system represented by reaction (3.17). The hydroperoxy radical HO2 is considered to be relatively unreactive so that it is able to diffuse to the wall and thus become a means for effectively destroying H radicals. The upper (third) explosion limit is due to a reaction that overtakes the stability of the HO2 and is possibly the sequence HO2  H 2 → H 2 O2  H ↓ 2OH

(3.22)

The reactivity of HO2 is much lower than that of OH, H, or O; therefore, somewhat higher temperatures are necessary for sequence [Eq. (3.22)] to become effective [6a]. Water vapor tends to inhibit explosion due to the effect of reaction (3.21) in that H2O has a high third-body efficiency, which is most probably due to some resonance energy exchange with the HO2 formed. Since reaction (3.21) is a recombination step requiring a third body, its rate decreases with increasing temperature, whereas the rate of reaction (3.17) increases with temperature. One then can generally conclude that reaction (3.17) will dominate at higher temperatures and lower pressures, while reaction (3.21) will be more effective at higher pressures and lower temperatures. Thus, in order to explain the limits in Fig. 3.2 it becomes apparent that at temperatures above 875 K, reaction (3.17) always prevails and the mixture is explosive for the complete pressure range covered.

88

Combustion

1400 H  O2 → OH  O 1300 Temperature (K)

1200 1100

10 1

1000

0.1

900 800

H  O2  M → HO2  M

700 600 500 400 0.1

1

10 Pressure (atm)

100

FIGURE 3.3 Ratio of the rates of H  O2 → OH  O to H  O2  M → HO2 M at various total pressures.

In this higher temperature regime and in atmospheric-pressure flames, the eventual fate of the radicals formed is dictated by recombination. The principal gas-phase termination steps are H  H  M → H2  M

(3.23)

O  O  M → O2  M

(3.24)

H  O  M → OH  M

(3.25)

H  OH  M → H 2 O  M

(3.26)

In combustion systems other than those whose lower-temperature explosion characteristics are represented in Fig. 3.2, there are usually ranges of temperature and pressure in which the rates of reactions (3.17) and (3.21) are comparable. This condition can be specified by the simple ratio k17 1 k21 (M) Indeed, in developing complete mechanisms for the oxidation of CO and hydrocarbons applicable to practical systems over a wide range of temperatures and high pressures, it is important to examine the effect of the HO2 reactions when the ratio is as high as 10 or as low as 0.1. Considering that for air combustion the total concentration (M) can be that of nitrogen, the boundaries of this ratio are depicted in Fig. 3.3, as derived from the data in Appendix C. These modern rate data indicate that the second explosion limit, as determined

89

Explosive and General Oxidative Characteristics of Fuels

10 Third limit 1

“Extended” second limit

P/atm

No explosion 0.1 Second limit 0.01

Explosion First limit

0.001 600

700

800

900 T/K

1000 1100

1200

FIGURE 3.4 The extended second explosion limit of H2¶O2 (after Ref. [6a]).

by glass vessel experiments and many other experimental configurations, as shown in Fig. 3.2, has been extended (Fig. 3.4) and verified experimentally [6a]. Thus, to be complete for the H2¶O2 system and other oxidation systems containing hydrogen species, one must also consider reactions of HO2. Sometimes HO2 is called a metastable species because it is relatively unreactive as a radical. Its concentrations can build up in a reacting system. Thus, HO2 may be consumed in the H2¶O2 system by various radicals according to the following reactions [4]: HO2  H → H 2  O2

(3.27)

HO2  H → OH  OH

(3.28)

HO2  H → H 2 O  O

(3.29)

HO2  O → O2  OH

(3.30)

HO2  OH → H 2 O  O2

(3.30a)

The recombination of HO2 radicals by HO2  HO2 → H 2 O2  O2

(3.31)

yields hydrogen peroxide (H2O2), which is consumed by reactions with radicals and by thermal decomposition according to the following sequence: H 2 O2  OH → H 2 O  HO2

(3.32)

90

Combustion

H 2 O2  H → H 2 O  OH

(3.33)

H 2 O2  H → HO2  H 2

(3.34)

H 2 O2  M → 2OH  M

(3.35)

From the sequence of reactions (3.32)–(3.35) one finds that although reaction (3.21) terminates the chain under some conditions, under other conditions it is part of a chain propagating path consisting essentially of reactions (3.21) and (3.28) or reactions (3.21), (3.31), and (3.35). It is also interesting to note that, as are most HO2 reactions, these two sequences of reactions are very exothermic; that is, H  O2  M → HO2  M HO2  H → 2OH 2H  O2 → 2OH  350 kJ/mol and H  O2  M → HO2  M HO2  HO2 → H 2 O2  O2 H 2 O2  M → 2OH  M H  HO2 → 2OH  156 kJ/mol Hence they can significantly affect the temperature of an (adiabatic) system and thereby move the system into an explosive regime. The point to be emphasized is that slow competing reactions can become important if they are very exothermic. It is apparent that the fate of the H atom (radical) is crucial in determining the rate of the H2¶O2 reaction or, for that matter, the rate of any hydrocarbon oxidation mechanism. From the data in Appendix C one observes that at temperatures encountered in flames the rates of reaction between H atoms and many hydrocarbon species are considerably larger than the rate of the chain branching reaction (3.17). Note the comparisons in Table 3.1. Thus, these reactions compete very effectively with reaction (3.17) for H atoms and reduce the chain branching rate. For this reason, hydrocarbons act as inhibitors for the H2¶O2 system [4]. As implied, at highly elevated pressures (P  20 atm) and relatively low temperatures (T  1000 K), reaction (3.21) will dominate over reaction (3.17); and as shown, the sequence of reactions (3.21), (3.31), and (3.35) provides the chain propagation. Also, at higher temperatures, when H  O2 → OH  O is microscopically balanced, reaction (3.21) (H  O2  M → ΗO2  M)

91

Explosive and General Oxidative Characteristics of Fuels

TABLE 3.1 Rate Constants of Specific Radical Reactions Rate constant

1000 K

2000 K

k (C3H8  OH)

5.0  10

12

1.6  1013

k (H2  OH)

1.6  1012

6.0  1012

k (CO  OH)

1.7  1011

3.5  1011

k (H  C3H8) → iC3H7

7.1  1011

9.9  1012

k (H  O2)

4.7  1010

3.2  1012

can compete favorably with reaction (3.17) for H atoms since the net removal of H atoms from the system by reaction (3.17) may be small due to its equilibration. In contrast, when reaction (3.21) is followed by the reaction of the fuel with HO2 to form a radical and hydrogen peroxide and then by reaction (3.35), the result is chain branching. Therefore, under these conditions increased fuel will accelerate the overall rate of reaction and will act as an inhibitor at lower pressures due to competition with reaction (3.17) [4]. The detailed rate constants for all the reactions discussed in this section are given in Appendix C. The complete mechanism for CO or any hydrocarbon or hydrogen-containing species should contain the appropriate reactions of the H2¶O2 steps listed in Appendix C; one can ignore the reactions containing Ar as a collision partner in real systems. It is important to understand that, depending on the temperature and pressure of concern, one need not necessarily include all the H2¶O2 reactions. It should be realized as well that each of these reactions is a set comprising a forward and a backward reaction; but, as the reactions are written, many of the backward reactions can be ignored. Recall that the backward rate constant can be determined from the forward rate constant and the equilibrium constant for the reaction system.

D. EXPLOSION LIMITS AND OXIDATION CHARACTERISTICS OF CARBON MONOXIDE Early experimental work on the oxidation of carbon monoxide was confused by the presence of any hydrogen-containing impurity. The rate of CO oxidation in the presence of species such as water is substantially faster than the “bone-dry” condition. It is very important to realize that very small quantities of hydrogen, even of the order of 20 ppm, will increase the rate of CO oxidation substantially [8]. Generally, the mechanism with hydrogen-containing compounds present is referred to as the “wet” carbon monoxide condition.

92

Combustion

700 Pressure (mm Hg)

600 500

Steady reaction

400 300 Explosion

200 100 0 560

600

640

680

720

760

Temperature (C) FIGURE 3.5

Explosion limits of a CO¶O2 mixture (after Ref. [2]).

Obviously, CO oxidation will proceed through this so-called wet route in most practical systems. It is informative, however, to consider the possible mechanisms for dry CO oxidation. Again the approach is to consider the explosion limits of a stoichiometric, dry CO¶O2 mixture. However, neither the explosion limits nor the reproducibility of these limits is well defined, principally because the extent of dryness in the various experiments determining the limits may not be the same. Thus, typical results for explosion limits for dry CO would be as depicted in Fig. 3.5. Figure 3.5 reveals that the low-pressure ignition of CO¶O2 is characterized by an explosion peninsula very much like that in the case of H2¶O2. Outside this peninsula one often observes a pale-blue glow, whose limits can be determined as well. A third limit has not been defined; and, if it exists, it lies well above 1 atm. As in the case of H2¶O2 limits, certain general characteristics of the defining curve in Fig. 3.5 may be stated. The lower limit meets all the requirements of wall destruction of a chain propagating species. The effects of vessel diameter, surface character, and condition have been well established by experiment [2]. Under dry conditions the chain initiating step is CO  O2 → CO2  O

(3.36)

which is mildly exothermic, but slow at combustion temperatures. The succeeding steps in this oxidation process involve O atoms, but the exact nature of these steps is not fully established. Lewis and von Elbe [2] suggested that chain branching would come about from the step O  O 2  M → O3  M

(3.37)

This reaction is slow, but could build up in supply. Ozone (O3) is the metastable species in the process (like HO2 in H2¶O2 explosions) and could initiate

Explosive and General Oxidative Characteristics of Fuels

93

chain branching, thus explaining the explosion limits. The branching arises from the reaction O3  CO → CO2  2O

(3.38)

Ozone destruction at the wall to form oxygen molecules would explain the lower limit. Lewis and von Elbe explain the upper limit by the third-order reaction O3  CO  M → CO2  O2  M

(3.39)

However, O3 does not appear to react with CO below 523 K. Since CO is apparently oxidized by the oxygen atoms formed by the decomposition of ozone [the reverse of reaction (3.37)], the reaction must have a high activation energy ( 120 kJ/mol). This oxidation of CO by O atoms was thought to be rapid in the high-temperature range, but one must recall that it is a three-body recombination reaction. Analysis of the glow and emission spectra of the CO¶O2 reaction suggests that excited carbon dioxide molecules could be present. If it is argued that O atoms cannot react with oxygen (to form ozone), then they must react with the CO. A suggestion of Semenov was developed further by Gordon and Knipe [9], who gave the following alternative scheme for chain branching: CO  O → CO*2

(3.40)

CO*2  O2 → CO2  2O

(3.41)

where CO*2 is the excited molecule from which the glow appears. This process is exothermic and might be expected to occur. Gordon and Knipe counter the objection that CO*2 is short-lived by arguing that through system crossing in excited states its lifetime may be sufficient to sustain the process. In this scheme the competitive three-body reaction to explain the upper limit is the aforementioned one: CO  O  M → CO2  M

(3.42)

Because these mechanisms did not explain shock tube rate data, Brokaw [8] proposed that the mechanism consists of reaction (3.36) as the initiation step with subsequent large energy release through the three-body reaction (3.42) and O  O  M → O2  M

(3.43)

The rates of reactions (3.36), (3.42), and (3.43) are very small at combustion temperatures, so that the oxidation of CO in the absence of any

94

Combustion

hydrogen-containing material is very slow. Indeed it is extremely difficult to ignite and have a flame propagate through a bone-dry, impurity-free CO¶O2 mixture. Very early, from the analysis of ignition, flame speed, and detonation velocity data, investigators realized that small concentrations of hydrogencontaining materials would appreciably catalyze the kinetics of CO¶O2. The H2O-catalyzed reaction essentially proceeds in the following manner: CO  O2 → CO2  O

(3.36)

O  H 2 O → 2OH

(3.20)

CO  OH → CO2  H

(3.44)

H  O2 → OH  O

(3.17)

O  H 2 → OH  H

(3.18)

OH  H 2 → H 2 O  H

(3.19)

If H2 is the catalyst, the steps

should be included. It is evident then that all of the steps of the H2¶O2 reaction scheme should be included in the so-called wet mechanism of CO oxidation. As discussed in the previous section, the reaction H  O2  M → HO2  M

(3.21)

enters and provides another route for the conversion of CO to CO2 by CO  HO2 → CO2  OH

(3.45)

At high pressures or in the initial stages of hydrocarbon oxidation, high concentrations of HO2 can make reaction (3.45) competitive to reaction (3.44), so reaction (3.45) is rarely as important as reaction (3.44) in most combustion situations [4]. Nevertheless, any complete mechanism for wet CO oxidation must contain all the H2¶O2 reaction steps. Again, a complete mechanism means both the forward and backward reactions of the appropriate reactions in Appendix C. In developing an understanding of hydrocarbon oxidation, it is important to realize that any high-temperature hydrocarbon mechanism involves H2 and CO oxidation kinetics, and that most, if not all, of the CO2 that is formed results from reaction (3.44).

95

Explosive and General Oxidative Characteristics of Fuels

≠

k CO  OH (cm3/mol/s)

1012

1011

1010

0

1

2

3

1000/T (K1) FIGURE 3.6 Reaction rate constant of the CO  OH reaction as a function of the reciprocal temperature based on transition state (—) and Arrhenius (--) theories compared with experimental data (after Ref. [10]).

The very important reaction (3.44) actually proceeds through a four-atom activated complex [10, 11] and is not a simple reaction step like reaction (3.17). As shown in Fig. 3.6, the Arrhenius plot exhibits curvature [10]. And because the reaction proceeds through an activated complex, the reaction rate exhibits some pressure dependence [12]. Just as the fate of H radicals is crucial in determining the rate of the H2¶O2 reaction sequence in any hydrogen-containing combustion system, the concentration of hydroxyl radicals is also important in the rate of CO oxidation. Again, as in the H2¶O2 reaction, the rate data reveal that reaction (3.44) is slower than the reaction between hydroxyl radicals and typical hydrocarbon species; thus one can conclude—correctly—that hydrocarbons inhibit the oxidation of CO (see Table 3.1). It is apparent that in any hydrocarbon oxidation process CO is the primary product and forms in substantial amounts. However, substantial experimental evidence indicates that the oxidation of CO to CO2 comes late in the reaction scheme [13]. The conversion to CO2 is retarded until all the original fuel and intermediate hydrocarbon fragments have been consumed [4, 13]. When these species have disappeared, the hydroxyl concentration rises to high levels and converts CO to CO2. Further examination of Fig. 3.6 reveals that the rate of reaction (3.44) does not begin to rise appreciably until the reaction reaches temperatures above 1100 K. Thus, in practical hydrocarbon combustion systems whose temperatures are of the order of 1100 K and below, the complete conversion of CO to CO2 may not take place. As an illustration of the kinetics of wet CO oxidation, Fig. 3.7 shows the species profiles for a small amount of CO reacting in a bath of O2 and H2O at constant

96

Combustion

0.01 CO2

CO 104

O2

HO2 H2 OH

O

Yi

106

H2O

108 H 1010 H2O2 1012 0.0001 0.001

0.01

0.1 Time (s)

1

10

100

FIGURE 3.7 Species mass fraction profiles for a constant temperature reaction of moist CO oxidation. Initial conditions: temperature  1100 K, pressure  1 atm, XCO  0.002, XH2O  0.01, XO2  0.028, and the balance N2 where Xi are the initial mole fractions.

temperature and constant pressure. The governing equations were described previously in Chapter 2. The induction period, during which the radical pool is formed and reaches superequilibrium concentrations lasts for approximately 5 ms. Shortly after the CO starts to react, the radicals obtain their maximum concentrations and are then consumed with CO until thermodynamic equilibrium is reached approximately 30 s later. However, 90% of the CO is consumed in about 80 ms. As one might expect, for these fuel lean conditions and a temperature of 1100 K, the OH and O intermediates are the most abundant radicals. Also note that for CO oxidation, as well as H2 oxidation, the induction times and ignition times are the same. Whereas the induction time describes the early radical pool growth and the beginning of fuel consumption, the ignition time describes the time for onset of significant heat release. It will be shown later in this chapter that for hydrocarbon oxidation the two times are generally different. Solution of the associated sensitivity analysis equations (Fig. 3.8) gives the normalized linear sensitivity coefficients for the CO mass fraction with respect to various rate constants. A rank ordering of the most important reactions in decreasing order is CO  OH → CO2  H

(3.44f)

97

Explosive and General Oxidative Characteristics of Fuels

4 21f

17b

44b

2 35b

20b

∂ ln YCO/∂ ln kj

0 30af

35f

2

20f

17f

4

44f

6

8 0.0001

0.001

0.01

0.1 Time (s)

1

10

100

FIGURE 3.8 Normalized first-order elementary sensitivity coefficients of the CO mass fraction with respect to various reaction rate constants. The arrows connect the subscript j with the corresponding profile of the sensitivity coefficient.

H  O2  M → HO2  M

(3.21f)

H  O2 → OH  O

(3.17f)

O  OH → H  O2

(3.17b)

O  H 2 O → OH  OH

(3.20f)

OH  OH → O  H 2 O

(3.20b)

The reverse of reaction (3.44) has no effect until the system has equilibrated, at which point the two coefficients ∂ ln YCO/∂ ln k44f and ∂ ln YCO/∂ ln k44b are equal in magnitude and opposite in sense. At equilibrium, these reactions are microscopically balanced, and therefore the net effect of perturbing both rate constants simultaneously and equally is zero. However, a perturbation of the ratio (k44f/k44b  K44) has the largest effect of any parameter on the CO equilibrium concentration. A similar analysis shows reactions (3.17) and (3.20) to become balanced shortly after the induction period. A reaction flux (rate-ofproduction) analysis would reveal the same trends.

98

Combustion

E. EXPLOSION LIMITS AND OXIDATION CHARACTERISTICS OF HYDROCARBONS To establish the importance of the high-temperature chain mechanism through the H2¶O2 sequence, the oxidation of H2 was discussed in detail. Also, because CO conversion to CO2 is the highly exothermic portion of any hydrocarbon oxidation system, CO oxidation was then detailed. Since it will be shown that all carbon atoms in alkyl hydrocarbons and most in aromatics are converted to CO through the radical of formaldehyde (H2CO) called formyl (HCO), the oxidation of aldehydes will be the next species to be considered. Then the sequence of oxidation reactions of the C1 to C5 alkyl hydrocarbons is considered. These systems provide the backdrop for consideration of the oxidation of the hydrocarbon oxygenates—alcohols, ether, ketenes, etc. Finally, the oxidation of the highly stabilized aromatics will be analyzed. This hierarchical approach should facilitate the understanding of the oxidation of most hydrocarbon fuels. The approach is to start with analysis of the smallest of the hydrocarbon molecules, methane. It is interesting that the combustion mechanism of methane was for a long period of time the least understood. In recent years, however, there have been many studies of methane, so that to a large degree its specific oxidation mechanisms are known over various ranges of temperatures. Now among the best understood, these mechanisms will be detailed later in this chapter. The higher-order hydrocarbons, particularly propane and above, oxidize much more slowly than hydrogen and are known to form metastable molecules that are important in explaining the explosion limits of hydrogen and carbon monoxide. The existence of these metastable molecules makes it possible to explain qualitatively the unique explosion limits of the complex hydrocarbons and to gain some insights into what the oxidation mechanisms are likely to be. Mixtures of hydrocarbons and oxygen react very slowly at temperatures below 200°C; as the temperature increases, a variety of oxygen-containing compounds can begin to form. As the temperature is increased further, CO and H2O begin to predominate in the products and H2O2 (hydrogen peroxide), CH2O (formaldehyde), CO2, and other compounds begin to appear. At 300–400°C, a faint light often appears, and this light may be followed by one or more blue flames that successively traverse the reaction vessel. These light emissions are called cool flames and can be followed by an explosion. Generally, the presence of aldehydes is revealed. In discussing the mechanisms of hydrocarbon oxidation and, later, in reviewing the chemical reactions in photochemical smog, it becomes necessary to identify compounds whose structure and nomenclature may seem complicated to those not familiar with organic chemistry. One need not have a background in organic chemistry to follow the combustion mechanisms; one should, however, study the following section to obtain an elementary knowledge of organic nomenclature and structure.

99

Explosive and General Oxidative Characteristics of Fuels

1. Organic Nomenclature No attempt is made to cover all the complex organic compounds that exist. The classes of organic compounds reviewed are those that occur most frequently in combustion processes and photochemical smog.

a. Alkyl Compounds

Paraffins (alkanes: single bonds)

C

CH4, C2H6, C3H8, C4H10,.., CnH2n2 Methane, ethane, propane, butane,..., straight-chain; isobutane, branched chain All are saturated (i.e., no more hydrogen can be added to any of the compounds) Radicals deficient in one H atom take the names methyl, ethyl, propyl, etc.

C

Olefins (alkenes: contain double bonds)

C

C2H4, C3H6, C4H8, …, CnH2n Ethene, propene, butane (ethylene, propylene, butylene) Diolefins contain two double bonds The compounds are unsaturated since CnH2n can be saturated to CnH2n2

C

CnH2n- no double bonds Cyclopropane, cyclobutane, cyclopentane Compounds are unsaturated since ring can be broken CnH2n  H2 → CnH2n2

Cycloparaffins (cycloalkanes: single bonds)

C

C C

Acetylenes (alkynes: contain triple bonds) C

C2H2, C3H4, C4H6, …, CnH2n2 Ethyne, propyne, butyne (acetylene, methyl acetylene, ethyl acetylene) Unsaturated compounds

C

b. Aromatic Compounds The building block for the aromatics is the ring-structured benzene C6H6, which has many resonance structures and is therefore very stable: H C HC HC C H

CH

HC

CH

HC C H

H C

H C

H C CH

HC

CH

HC C H

CH

HC

CH

HC C H

H C CH

HC

CH

HC

CH CH C H

100

Combustion

The ring structure of benzene is written in shorthand as either

or φΗ

where φ is the phenyl radical: C6H5. Thus CH3

CH3

OH

CH3

Toluene or φCH3

Phenol (benzol)

Xylene

or φOH

xylene being ortho, meta, or para according to whether methyl groups are separated by one, two, or three carbon atoms, respectively. Polyaromatic hydrocarbons (PAH) are those which exists as combined aromatic ring structures represented by naphthalene (C10H8); H C

H C HC

CH

C

HC

CH

C C H

C H

α-methylnaphthalene has a methyl radical attachment at one of the peak carbon atoms. If β is used then the methyl radical is attached to one of the other non-associated carbon atoms.

c. Alcohols Those organic compounds that contain a hydroxyl group (-OH) are called alcohols and follow the simple naming procedure. CH3OH

C2H5OH

Methanol (methyl alcohol)

Ethanol (ethyl alcohol)

101

Explosive and General Oxidative Characteristics of Fuels

The bonding arrangement is always

C

OH

d. Aldehydes The aldehydes contain the characteristic formyl radical group O C H

and can be written as O R

C H

where R can be a hydrogen atom or an organic radical. Thus O

O H

C

H Formaldehyde

H3C

O

C

H5C2

H Acetaldehyde

H Proprionaldehyde

e. Ketones The ketones contain the characteristic group O C

and can be written more generally as O R

C

C

R'

102

Combustion

where R and R are always organic radicals. Thus O H5C2

C

CH3

is methyl ethyl ketone.

f. Organic Acids Organic acids contain the group O C OH

and are generally written as O R

C OH

where R can be a hydrogen atom or an organic radical. Thus O C

H

O H3C

C

OH Formic acid

OH Acetic acid

g. Organic Salts O R

C

O H3C

OONO2 Peroxyacyl nitrate

C OONO2

Peroxyacetyl nitrate

103

Explosive and General Oxidative Characteristics of Fuels

h. Others Ethers take the form R¶O¶R, where R and R are organic radicals. The peroxides take the form R¶O¶O¶R or R¶O¶O¶H, in which case the term hydroperoxide is used.

2. Explosion Limits At temperatures around 300–400°C and slightly higher, explosive reactions in hydrocarbon–air mixtures can take place. Thus, explosion limits exist in hydrocarbon oxidation. A general representation of the explosion limits of hydrocarbons is shown in Fig. 3.9. The shift of curves, as shown in Fig. 3.9, is unsurprising since the larger fuel molecules and their intermediates tend to break down more readily to form radicals that initiate fast reactions. The shape of the propane curve suggests that branched chain mechanisms are possible for hydrocarbons. One can conclude that the character of the propane mechanism is different from that of the H2¶O2 reaction when one compares this explosion curve with the H2¶O2 pressure peninsula. The island in the propane–air curve drops and goes slightly to the left for higher-order paraffins; for example, for hexane it occurs at 1 atm. For the reaction of propane with pure oxygen, the curve drops to about 0.5 atm. Hydrocarbons exhibit certain experimental combustion characteristics that are consistent both with the explosion limit curves and with practical considerations; these characteristics are worth reviewing: ●

Pressure (atm)

●

Hydrocarbons exhibit induction intervals that are followed by a very rapid reaction rate. Below 400°C, these rates are of the order of 1 s or a fraction thereof, and below 300°C they are of the order of 60 s. Their rate of reaction is inhibited strongly by adding surface (therefore, an important part of the reaction mechanism must be of the free-radical type).

Explosion

4 2

1 2

3

Methane

4

Ethane

Cool flames

Steady reaction 300

Propane

400 Temperature (C)

FIGURE 3.9 General explosion limit characteristics of stoichiometric hydrocarbon–air mixture. The dashed box denotes cool flame region.

104

●

● ● ●

●

Combustion

They form aldehyde groups, which appear to have an influence (formaldehyde is the strongest). These groups accelerate and shorten the ignition lags. They exhibit cool flames, except in the cases of methane and ethane. They exhibit negative temperature coefficients of reaction rate. They exhibit two-stage ignition, which may be related to the cool flame phenomenon. Their reactions are explosive without appreciable self-heating (branched chain explosion without steady temperature rise). Explosion usually occurs when passing from region 1 to region 2 in Fig. 3.9. Explosions may occur in other regions as well, but the reactions are so fast that we cannot tell whether they are self-heating or not.

a. The Negative Coefficient of Reaction Rate Semenov [14] explained the long induction period by hypothesizing unstable, but long-lived species that form as intermediates and then undergo different reactions according to the temperature. This concept can be represented in the form of the following competing fuel (A) reaction routes after the formation of the unstable intermediate M*: I (non-chain branching step) A

M* II (chain branching step)

Route I is controlled by an activation energy process larger than that of II. Figure 3.10 shows the variation of the reaction rate of each step as a function of temperature. The numbers in Fig. 3.10 correspond to the temperature position designation in Fig. 3.9. At point 1 in Fig. 3.10 one has a chain branching system since the temperature is low and αcrit is large; thus, α αcrit and the system is nonexplosive. As the temperature is increased (point 2), the rate constants of the chain steps in the system increase and αcrit drops; so α αcrit and the system explodes. At a still higher temperature (point 3), the non-chain branching route I becomes faster. Although this step is faster, α is always less than αcrit; thus the system cannot explode. Raising temperatures along route I still further leads to a reaction so fast that it becomes self-heating and hence explosive again (point 4). The temperature domination explains the peninsula in the P–T diagram (Fig. 3.9), and the negative coefficient of reaction rate is due to the shift from point 2 to 3.

b. Cool Flames The cool-flame phenomenon [15] is generally a result of the type of experiment performed to determine the explosion limits and the negative temperature coefficient feature of the explosion limits. The chemical mechanisms used to explain these phenomena are now usually referred to as cool-flame chemistry.

105

Explosive and General Oxidative Characteristics of Fuels

4

In k

I

3 II 2 1

1/T FIGURE 3.10 Arrhenius plot of the Semenov steps in hydrocarbon oxidation. Points 1–4 correspond to the same points as in Fig. 3.9.

Most explosion limit experiments are performed in vessels immersed in isothermal liquid baths (see Fig. 3.1). Such systems are considered to be isothermal within the vessel itself. However, the cool gases that must enter will become hotter at the walls than in the center. The reaction starts at the walls and then propagates to the center of the vessel. The initial reaction volume, which is the hypothetical outermost shell of gases in the vessel, reaches an explosive condition (point 2). However, owing to the exothermicity of the reaction, the shell’s temperature rises and moves the reacting system to the steady condition point 3; and because the reaction is slow at this condition, not all the reactants are consumed. Each successive inner (shell) zone is initiated by the previous zone and progresses through the steady reaction phase in the same manner. Since some chemiluminescence occurs during the initial reaction stages, it appears as if a flame propagates through the mixture. Indeed, the events that occur meet all the requirements of an ordinary flame, except that the reacting mixture loses its explosive characteristic. Thus there is no chance for the mixture to react completely and reach its adiabatic flame temperature. The reactions in the system are exothermic and the temperatures are known to rise about 200°C—hence the name “cool flames.” After the complete vessel moves into the slightly higher temperature zone, it begins to be cooled by the liquid bath. The mixture temperature drops, the system at the wall can move into the explosive regime again, and the phenomenon can repeat itself since all the reactants have not been consumed. Depending on the specific experimental conditions and mixtures under study, as many as five cool flames have been known to propagate through a given single mixture. Cool flames have been observed in flow systems, as well [16].

106

Combustion

3. “Low-Temperature” Hydrocarbon Oxidation Mechanisms It is essential to establish the specific mechanisms that explain the cool flame phenomenon, as well as the hydrocarbon combustion characteristics mentioned earlier. Semenov [14] was the first to propose the general mechanism that formed the basis of later research, which clarified the processes taking place. This mechanism is written as follows:   HO RH  O2 → R 2  R  O2 → olefin  HO 2  R  O2 → RO 2   RH → ROOH  R  RO 2  → RCHO  R O  RO 2   RH → H O  R HO 2 2 2   OH  ROOH → RO



(initiation)

(3.47) (3.48) (chain propagating)

(degenerate branching)

  HO  RCHO  O2 → RCO 2  → destruction RO 2

(3.46)

(chain terminating)

(3.49) (3.50) (3.51) (3.52) (3.53) (3.54)

where the dot above a particular atom designates the radical position. This scheme is sufficient for all hydrocarbons with a few carbon atoms, but for multicarbon ( 5) species, other intermediate steps must be added, as will be shown later. Since the system requires the buildup of ROOH and RCHO before chain branching occurs to a sufficient degree to dominate the system, Semenov termed these steps degenerate branching. This buildup time, indeed, appears to account for the experimental induction times noted in hydrocarbon combustion systems. It is important to emphasize that this mechanism is a low-temperature scheme and consequently does not include the high-temperature H2¶O2 chain branching steps. At first, the question of the relative importance of ROOH versus aldehydes as intermediates was much debated; however, recent work indicates that the hydroperoxide step dominates. Aldehydes are quite important as fuels in the cool-flame region, but they do not lead to the important degenerate chain  compounds form ROH species, which play branching step as readily. The RO no role with respect to the branching of concern.

Explosive and General Oxidative Characteristics of Fuels

107

Owing to its high endothermicity, the chain initiating reaction is not an important route to formation of the radical R once the reaction system has created other radicals. Obviously, the important generation step is a radical attack on the fuel, and the fastest rate of attack is by the hydroxyl radicals since this reaction step is highly exothermic owing to the creation of water as a product. So the system for obtaining R comes from the reactions  →R   XH RH  X

(3.55)

 →R   HOH RH  OH

(3.56)

where X represents any radical. It is the fate of the hydrocarbon radical that determines the existence of the negative temperature coefficient and cool  forms via reaction (3.48). The structure flames. The alkyl peroxy radical RO 2 of this radical can be quite important. The H abstracted from RH to form the radical R comes from a preferential position. The weakest C¶H bonding is on a tertiary carbon; and, if such C atoms exist, the O2 will preferentially attack this position. If no tertiary carbon atoms exist, the preferential attack occurs on the next weakest C¶H bonds, which are those on the second carbon atoms from the ends of the chain (refer to Appendix D for all bond strengths). Then,  becomes the predominant attacker of as the hydroxyl radical pool builds, OH the fuel. Because of the energetics of the hydroxyl step (56), for all intents and purposes, it is relatively nonselective in hydrogen abstraction. It is known that when O2 attaches to the radical, it forms a near 90° angle with the carbon atoms (the realization of this stearic condition will facilitate understanding of certain reactions to be depicted later). The peroxy radical abstracts a H from any fuel molecule or other hydrogen donor to form the hydroperoxide (ROOH) [reaction (3.49)]. Tracing the steps, one realizes that the amount of hydroperoxy radical that will form depends on the competition of reaction (3.48) with reaction (3.47), which forms the stable olefin together  that forms from reaction (3.47) then forms the peroxide  . The HO with HO 2 2 H2O2 through reaction (3.51). At high temperatures H2O2 dissociates into two hydroxyl radicals; however, at the temperatures of concern here, this dissociation does not occur and the fate of the H2O2 (usually heterogeneous) is to form water and oxygen. Thus, reaction (3.47) essentially leads only to steady reaction. In brief, then, under low-temperature conditions it is the competition between reactions (3.47) and (3.48) that determines whether the fuel–air mixture will become explosive or not. Its capacity to explode depends on whether the chain system formed is sufficiently branching to have an α greater than αcrit.

a. Competition between Chain Branching and Steady Reaction Steps Whether the sequence given as reactions (3.46)–(3.54) becomes chain branching or not depends on the competition between the reactions    O → olefin  HO R 2 2

(3.47)

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Combustion

and    O → RO R 2 2

(3.48)

Some evidence [17, 17a] suggests that both sets of products develop from a complex via a process that can be written as  * → R O H* → olefin  HO    O  RO R 2 2 H 2 2 ↓ [M]  RO 2

(3.57)

At low temperatures and modest pressures, a significant fraction of the complex dissociates back to reactants. A small fraction of the complex at low pressures then undergoes the isomerization  * → R O H* RO 2 −H 2

(3.58)

and subsequent dissociation to the olefin and HO2. Another small fraction is  : stabilized to form RO 2 [M]  * ⎯ ⎯⎯  RO → RO 2 2

(3.59)

With increasing pressure, the fraction of the activated complex that is stabilized will approach unity [17]. As the temperature increases, the route to the olefin becomes favored. The direct abstraction leading to the olefin reaction (3.47) must therefore become important at some temperature higher than 1000 K [17a].

b. Importance of Isomerization in Large Hydrocarbon Radicals With large hydrocarbon molecules an important isomerization reaction will occur. Benson [17b] has noted that with six or more carbon atoms, this reaction becomes a dominant feature in the chain mechanism. Since most practical fuels contain large paraffinic molecules, one can generalize the new competitive mechanisms as Isomerization

RH

X (a)

R

O2 (b)

RO2

ROOH

O2 (d)

branching chain

(c)

Decomposition (reverse of b)

Olefin—free radical straight chain

109

Explosive and General Oxidative Characteristics of Fuels

Note that the isomerization step is  → ROOH  RO 2

(3.61)

while the general sequence of step (d) is O2 RH I ¶ R II OOH ⎯ ⎯⎯⎯    ⎯ ⎯⎯ → OOR → HOOR I ¶ R II OOH  R ROOH IV III  →R CHO (3.62)   CO  R     2OH ketone

aldehyde

where the Roman numeral superscripts represent different hydrocarbon radicals R of smaller chain length than RH. It is this isomerization concept that requires one to add reactions to the Semenov mechanism to make this mechanism most general. The oxidation reactions of 2-methylpentane provide a good example of how the hydroperoxy states are formed and why molecular structure is important in establishing a mechanism. The C¶C bond angles in hydrocarbons are about 108°. The reaction scheme is then O

OOH

O H

H3C

C

H (3.61)

C

H3C

C

CH3

H

H

(3.61)

H3C

H C

C C

H3C

CH3

H

H

(3.63) O2

O

OOH

OH H

H2C

O

OH

C

C

CH3

C

H3C H

RH

(3.64)

H3C

H C

H3C

H

O C

CH3 H

H

CH3

OO C

C

O

CH2

CH3  CH3

 2OH

C

(3.65) H

Here one notices that the structure of the 90° (COO) bonding determines the intermediate ketone, aldehyde, and hydroxyl radicals that form.

110

Combustion

Although reaction (3.61) is endothermic and its reverse step reaction (–3.61) is faster, the competing step reaction (3.63) can be faster still; thus the isomerization [reaction (3.61)] step controls the overall rate of formation  and subsequent chain branching. This sequence essentially negates of ROO  and olefin the extent of reaction (–3.48). Thus the competition between ROO  production becomes more severe and it is more likely that ROO would form at the higher temperatures. It has been suggested [18] that the greater tendency for long-chain hydrocarbons to knock as compared to smaller and branched chain molecules may be a result of this internal, isomerization branching mechanism.

F. THE OXIDATION OF ALDEHYDES The low-temperature hydrocarbon oxidation mechanism discussed in the previous section is incomplete because the reactions leading to CO were not included. Water formation is primarily by reaction (3.56). The CO forms by  . The the conversion of aldehydes and their acetyl (and formyl) radicals, RCO same type of conversion takes place at high temperatures; thus, it is appropriate, prior to considering high-temperature hydrocarbon oxidation schemes, to develop an understanding of the aldehyde conversion process. As shown in Section E1, aldehydes have the structure O R

C H

 is the where R is either an organic radical or a hydrogen atom and HCO formyl radical. The initiation step for the high-temperature oxidation of aldehydes is the thermolysis reaction  HM RCHO  M → RCO

(3.66)

The CH bond in the formyl group is the weakest of all CH bonds in the molecule (see Appendix D) and is the one predominantly broken. The R¶C bond is substantially stronger than this CH bond, so cleavage of this bond as an initiation step need not be considered. As before, at lower temperatures, high pressures, and under lean conditions, the abstraction initiation step   HO RCHO  O2 → RCO 2

(3.53)

must be considered. Hydrogen-labeling studies have shown conclusively that the formyl H is the one abstracted—a finding consistent with the bond energies.

Explosive and General Oxidative Characteristics of Fuels

111

The H atom introduced by reaction (3.66) and the OH, which arises from the HO2, initiate the H radical pool that comes about from reactions (3.17)– (3.20). The subsequent decay of the aldehyde is then given by   XH RCHO  X → RCO

(3.67)

where X represents the dominant radicals OH, O, H, and CH3. The methyl radical CH3 is included not only because of its slow reactions with O2, but also because many methyl radicals are formed during the oxidation of practically all aliphatic hydrocarbons. The general effectiveness of each radical is in the order OH O H CH3, where the hydroxyl radical reacts the fastest with the aldehyde. In a general hydrocarbon oxidation system these radicals arise from steps other than reaction (3.66) for combustion processes, so the aldehyde oxidation process begins with reaction (3.67). An organic group R is physically much larger than an H atom, so the radical RCO is much more unstable than HCO, which would arise if R were a hydrogen atom. Thus one needs to consider only the decomposition of RCO in combustion systems; that is,   M → R  CO  M RCO

(3.68)

Similarly, HCO decomposes via HCO  M → H + CO + M

(3.69)

but under the usual conditions, the following abstraction reaction must play some small part in the process: HCO  O2 → CO  HO2

(3.70)

At high pressures the presence of the HO2 radical also contributes via HCO  HO2 → H2O2  CO, but HO2 is the least effective of OH, O, and H, as the rate constants in Appendix C will confirm. The formyl radical reacts very rapidly with the OH, O, and H radicals. However, radical concentrations are much lower than those of stable reactants and intermediates, and thus formyl reactions with these radicals are considered insignificant relative to the other formyl reactions. As will be seen when the oxidation of large hydrocarbon molecules is discussed (Section H), R is most likely a methyl radical, and the highest-order aldehydes to arise in high-temperature combustion are acetaldehyde and propionaldehyde. The acetaldehyde is the dominant form. Essentially, then, the sequence above was developed with the consideration that R was a methyl group.

112

Combustion

G. THE OXIDATION OF METHANE 1. Low-Temperature Mechanism Methane exhibits certain oxidation characteristics that are different from those of all other hydrocarbons. Tables of bond energy show that the first broken C¶H bond in methane takes about 40 kJ more than the others, and certainly more than the C¶H bonds in longer-chain hydrocarbons. Thus, it is not surprising to find various kinds of experimental evidence indicating that ignition is more difficult with methane/air (oxygen) mixtures than it is with other hydrocarbons. At low temperatures, even oxygen atom attack is slow. Indeed, in discussing exhaust emissions with respect to pollutants, the terms total hydrocarbons and reactive hydrocarbons are used. The difference between the two terms is simply methane, which reacts so slowly with oxygen atoms at atmospheric temperatures that it is considered unreactive. The simplest scheme that will explain the lower-temperature results of methane oxidation is the following:   CH 4  O2 → CH 3  HO 2 }   CH 3  O 2 → CH 2 O  OH   CH → H O  CH  OH 4 2 3   CH O → H O + HCO  OH 2 2



  HCO  } CH 2 O  O2 → HO 2   O → CO  HO  HCO 2 2   CH → H O  CH  HO 2 4 2 2 3   CH O → H O  HCO  HO 2 2 2 2  → wall OH CH 2 O → wall  → wall HO 2

 

(chain initiating)

(3.71) (3.72)

(chain propagating)

(3.73) (3.74)

(chain branching)

(3.75) (3.76)

(chain propagating)

(3.77) (3.78) (3.79)

(chain terminating)

(3.80) (3.81)

113

Explosive and General Oxidative Characteristics of Fuels

There is no H2O2 dissociation to OH radicals at low temperatures. H2O2 dissociation does not become effective until temperature reaches about 900 K. As before, reaction (3.71) is slow. Reactions (3.72) and (3.73) are faster since they involve a radical and one of the initial reactants. The same is true for reactions (3.75)–(3.77). Reaction (3.75) represents the necessary chain branching step. Reactions (3.74) and (3.78) introduce the formyl radical known to exist in the low-temperature combustion scheme. Carbon monoxide is formed by reaction (3.76), and water by reaction (3.73) and the subsequent decay of the peroxides formed. A conversion step of CO to CO2 is not considered because the rate of conversion by reaction (3.44) is too slow at the temperatures of concern here. It is important to examine more closely reaction (3.72), which proceeds [18, 19] through a metastable intermediate complex—the methyl peroxy radical—in the following manner:

CH3  O2

H

H

O

C

O

H

H

C

O  HO

(3.82)

H

At lower temperatures the equilibrium step is shifted strongly toward the complex, allowing the formaldehyde and hydroxyl radical formation. The structure of the complex represented in reaction (3.82) is well established. Recall that when O2 adds to the carbon atom in a hydrocarbon radical, it forms about a 90° bond angle. Perhaps more important, however, is the suggestion [18] that at temperatures of the order of 1000 K and above the equilibrium step in reaction (3.82) shifts strongly toward the reactants so that the overall reaction to form formaldehyde and hydroxyl cannot proceed. This condition would therefore pose a restriction on the rapid oxidation of methane at high temperatures. This possibility should come as no surprise as one knows that a particular reaction mechanism can change substantially as the temperature and pressure changes. There now appears to be evidence that another route to the aldehydes and OH formation by reaction (3.72) may be possible at high temperatures [6a, 19]; this route is discussed in the next section.

2. High-Temperature Mechanism Many extensive models of the high-temperature oxidation process of methane have been published [20, 20a, 20b, 21]. Such models are quite complex and include hundreds of reactions. The availability of sophisticated computers and computer programs such as those described in Appendix I permits the development of these models, which can be used to predict flow-reactor results, flame speeds, emissions, etc., and to compare these predictions with appropriate experimental data. Differences between model and experiment are used to modify the mechanisms and rate constants that are not firmly established. The purpose here is to point out the dominant reaction steps in these complex

114

Combustion

models of methane oxidation from a chemical point of view, just as modern sensitivity analysis [20, 20a, 20b] as shown earlier can be used to designate similar steps according to the particular application of the mechanism. The next section will deal with other, higher-order hydrocarbons. In contrast to reaction (3.71), at high temperatures the thermal decomposition of the methane provides the chain initiation step, namely CH 4  M → CH3  H  M

(3.83)

With the presence of H atoms at high temperature, the endothermic initiated H2¶O2 branching and propagating scheme proceeds, and a pool of OH, O, and H radicals develops. These radicals, together with HO2 [which would form if the temperature range were to permit reaction (3.71) as an initiating step], abstract hydrogen from CH4 according to CH 4  X → CH3  XH

(3.84)

where again X represents any of the radicals. The abstraction rates by the radicals OH, O, and H are all fast, with OH abstraction generally being the fastest. However, these reactions are known to exhibit substantial non-Arrhenius temperature behavior over the temperature range of interest in combustion. The rate of abstraction by O compared to H is usually somewhat faster, but the order could change according to the prevailing stoichiometry; that is, under fuel-rich conditions the H rate will be faster than the O rate owing to the much larger hydrogen atom concentrations under these conditions. The fact that reaction (3.82) may not proceed as written at high temperatures may explain why methane oxidation is slow relative to that of other hydrocarbon fuels and why substantial concentrations of ethane are found [4] during the methane oxidation process. The processes consuming methyl radicals are apparently slow, so the methyl concentration builds up and ethane forms through simple recombination: CH3  CH3 → C2 H6

(3.85)

Thus methyl radicals are consumed by other methyl radicals to form ethane, which must then be oxidized. The characteristics of the oxidation of ethane and the higher-order aliphatics are substantially different from those of methane (see Section H1). For this reason, methane should not be used to typify hydrocarbon oxidation processes in combustion experiments. Generally, a third body is not written for reaction (3.85) since the ethane molecule’s numerous internal degrees of freedom can redistribute the energy created by the formation of the new bond.

Explosive and General Oxidative Characteristics of Fuels

115

Brabbs and Brokaw [22] were among the first who suggested the main oxidation destruction path of methyl radicals to be  O  CH3  O2 → CH3 O

(3.86)

 is the methoxy radical. Reaction (3.86) is very endothermic and where CH3 O has a relatively large activation energy (120 kJ/mol [4]); thus it is quite slow for a chain step. There has been some question [23] as to whether reaction (3.72) could prevail even at high temperature, but reaction (3.86) is generally accepted as the major path of destruction of methyl radicals. Reaction (3.72) can be only a minor contribution at high temperatures. Other methyl radical reactions are [4] CH3  O → H 2 CO  H

(3.87)

CH3  OH → H 2 CO  H 2

(3.88)

CH3  OH → CH3 O  H

(3.89)

CH3  H 2 CO → CH 4  HCO

(3.90)

CH3  HCO → CH 4  CO

(3.91)

CH3  HO2 → CH3 O  OH

(3.92)

These are radical–radical reactions or reactions of methyl radicals with a product of a radical–radical reaction (owing to concentration effects) and are considered less important than reactions (3.72) and (3.86). However, reactions (3.72) and (3.86) are slow, and reaction (3.92) can become competitive to form the important methoxy radical, particularly at high pressures and in the lowertemperature region of flames (see Chapter 4). The methoxy radical formed by reaction (3.86) decomposes primarily and rapidly via CH3 O  M → H 2 CO  H  M

(3.93)

Although reactions with radicals to give formaldehyde and another product could be included, they would have only a very minor role. They have large rate constants, but concentration factors in reacting systems keep these rates slow. Reaction (3.86) is relatively slow for a chain branching step; nevertheless, it is followed by the very rapid decay reaction for the methoxy [reaction (3.93)],

116

Combustion

and the products of this two-step process are formaldehyde and two very reactive radicals, O and H. Similarly, reaction (3.92) may be equally important and can contribute a reactive OH radical. These radicals provide more chain branching than the low-temperature step represented by reaction (3.72), which produces formaldehyde and a single hydroxyl radical. The added chain branching from the reaction path [reactions (3.86) and (3.93)] may be what produces a reasonable overall oxidation rate for methane at high temperatures. In summary, the major reaction paths for the high-temperature oxidation of methane are CH 4  M → CH3  H  M

(3.83)

CH 4  X → CH3  XH

(3.84)

CH3  O2 → CH3 O  O

(3.86)

CH3  O2 → H 2 CO  OH

(3.72)

CH3 O  M → H 2 CO  H  M

(3.93)

H 2 CO  X → HCO  XH

(3.67)

HCO  M → H  CO  M

(3.69)

CH3  CH3 → C2 H6

(3.85)

CO  OH → CO2  H

(3.44)

Of course, all the appropriate higher-temperature reaction paths for H2 and CO discussed in the previous sections must be included. Again, note that when X is an H atom or OH radical, molecular hydrogen (H2) or water forms from reaction (3.84). As previously stated, the system is not complete because sufficient ethane forms so that its oxidation path must be a consideration. For example, in atmospheric-pressure methane–air flames, Warnatz [24, 25] has estimated that for lean stoichiometric systems about 30% of methyl radicals recombine to form ethane, and for fuel-rich systems the percentage can rise as high as 80%. Essentially, then, there are two parallel oxidation paths in the methane system: one via the oxidation of methyl radicals and the other via the oxidation of ethane. Again, it is worthy of note that reaction (3.84) with hydroxyl is faster than reaction (3.44), so that early in the methane system CO accumulates; later, when the CO concentration rises, it effectively competes with methane for hydroxyl radicals and the fuel consumption rate is slowed.

Explosive and General Oxidative Characteristics of Fuels

117

The mechanisms of CH4 oxidation covered in this section appear to be most appropriate, but are not necessarily definitive. Rate constants for various individual reactions could vary as the individual steps in the mechanism are studied further.

H. THE OXIDATION OF HIGHER-ORDER HYDROCARBONS 1. Aliphatic Hydrocarbons The high-temperature oxidation of paraffins larger than methane is a fairly complicated subject owing to the greater instability of the higher-order alkyl radicals and the great variety of minor species that can form (see Table 3.2). But, as is the case with methane [20, 20a, 20b, 21], there now exist detailed models of ethane [26], propane [27], and many other higher-order aliphatic hydrocarbons (see Cathonnet [28]). Despite these complications, it is possible to develop a general framework of important steps that elucidate this complex subject.

a. Overall View It is interesting to review a general pattern for oxidation of hydrocarbons in flames, as suggested very early by Fristrom and Westenberg [29]. They suggested two essential thermal zones: the primary zone, in which the initial hydrocarbons are attacked and reduced to products (CO, H2, H2O) and radicals (H, O, OH), and the secondary zone, in which CO and H2 are completely oxidized. The intermediates are said to form in the primary zone. Initially, then,

TABLE 3.2 Relative Importance of Intermediates in Hydrocarbon Combustion Fuel

Relative hydrocarbon intermediate concentrations

Ethane

ethene  methane

Propane

ethene propene  methane ethane

Butane

ethene propene  methane ethane

Hexane

ethene propene butene methane  pentene ethane

2-Methylpentane

propene ethene butene methane  pentene ethane

118

Combustion

hydrocarbons of lower order than the initial fuel appear to form in oxygenrich, saturated hydrocarbon flames according to

{ OH  Cn H2 n2

→ H 2 O  [Cn H 2 n1 } → Cn1H 2 n2  CH3 ]

Because hydrocarbon radicals of higher order than ethyl are unstable, the initial radical CnH2nl usually splits off CH3 and forms the next lower-order olefinic compound, as shown. With hydrocarbons of higher order than C3H8, there is fission into an olefinic compound and a lower-order radical. Alternatively, the radical splits off CH3. The formaldehyde that forms in the oxidation of the fuel and of the radicals is rapidly attacked in flames by O, H, and OH, so that formaldehyde is usually found only as a trace in flames. Fristrom and Westenberg claimed that the situation is more complex in fuel-rich saturated hydrocarbon flames, although the initial reaction is simply the H abstraction analogous to the preceding OH reaction; for example, H  Cn H 2 n2 → H 2  C2 H 2 n1 Under these conditions the concentrations of H and other radicals are large enough that their recombination becomes important, and hydrocarbons of order higher than the original fuel are formed as intermediates. The general features suggested by Fristrom and Westenberg were confirmed at Princeton [12, 30] by high-temperature flow-reactor studies. However, this work permits more detailed understanding of the high-temperature oxidation mechanism and shows that under oxygen-rich conditions the initial attack by O atoms must be considered as well as the primary OH attack. More importantly, however, it has been established that the paraffin reactants produce intermediate products that are primarily olefinic, and the fuel is consumed, to a major extent, before significant energy release occurs. The higher the initial temperature, the greater the energy release, as the fuel is being converted. This observation leads one to conclude that the olefin oxidation rate simply increases more appreciably with temperature; that is, the olefins are being oxidized while they are being formed from the fuel. Typical flow-reactor data for the oxidation of ethane and propane are shown in Figs. 3.11 and 3.12. The evidence in Figs. 3.11 and 3.12 [12, 30] indicates three distinct, but coupled zones in hydrocarbon combustion: 1. Following ignition, the primary fuel disappears with little or no energy release and produces unsaturated hydrocarbons and hydrogen. A little of the hydrogen is concurrently oxidized to water. 2. Subsequently, the unsaturated compounds are further oxidized to carbon monoxide and hydrogen. Simultaneously, the hydrogen present and formed is oxidized to water.

119

Explosive and General Oxidative Characteristics of Fuels

1.20 Total carbon

1250

Region 1 Region II Region III C2H6  2 CO2

0.80

1200 1150

0.60 1100

φ  0.092 1 cm  1.0 ms

0.40

C2H4  2 CO

Temp.

0.20

Temperature (K)

Mole percent species

1.00

1050 1000

0 0.02 0

H2 20

30

40

CH4  2

50 60 70 80 90 Distance from injection (cm)

100

110

120

FIGURE 3.11 Oxidation of ethane in a turbulent flow reactor showing intermediate and final product formation (after [13]). 0.80

Mole percent species

0.40 0.20

1150

Total carbon Temp. C3H8  3 C3H6  3 C2H4  2 CO2 CO Total carbon

Temp.

C3H8

1100 1050

CO CO2 C2H4

C3H6

Temperature (K)

0.60

1000

0 0.03 0.02

H2 CH4 C2H6  2 CH3 CHO  2

0.01 0 20

30

40

1 cm  1 ms   0.079

H2 CH4 CH3 CHO C2H6

50 60 70 80 90 Distance from injection (cm)

100

110

120

FIGURE 3.12 Oxidation of propane in a turbulent flow reactor (after [13]).

3. Finally, the large amounts of carbon monoxide formed are oxidized to carbon dioxide and most of the heat released from the overall reaction is obtained. Recall that the CO is not oxidized to CO2 until most of the fuel is consumed owing to the rapidity with which OH reacts with the fuel compared to its reaction to CO (see Table 3.1).

120

Combustion

b. Paraffin Oxidation In the high-temperature oxidation of large paraffin molecules, the chain initiation step is one in which a CC bond is broken to form hydrocarbon radicals; namely, RH (M) → R  R (M)

(3.94)

This step will undoubtedly dominate, since the CC bond is substantially weaker than any of the CH bonds in the molecule. As mentioned in the previous section, the radicals R and R (fragments of the original hydrocarbon molecule RH) decay into olefins and H atoms. At any reasonable combustion temperature, some CH bonds are broken and H atoms appear owing to the initiation step RH (M) → R  H (M)

(3.95)

For completeness, one could include a lower-temperature abstraction initiation step RH  O2 → R  HO2

(3.96)

The essential point is that the initiation steps provide H atoms that react with the oxygen in the system to begin the chain branching propagating sequence that nourishes the radical reservoir of OH, O, and H; that is, the reaction sequences for the complete H2¶O2 system must be included in any hightemperature hydrocarbon mechanism. Similarly, when CO forms, its reaction mechanism must be included as well. Once the radical pool forms, the disappearance of the fuel is controlled by the reactions RH  OH → R  H 2 O

(3.97)

RH  X → R  XH

(3.98)

where, again, X is any radical. For the high-temperature condition, X is primarily OH, O, H, and CH3. Since the RH under consideration is a multicarbon compound, the character of the radical R formed depends on which hydrogen in the molecule is abstracted. Furthermore, it is important to consider how the rate of reaction (3.98) varies as X varies, since the formation rates of the alkyl isomeric radicals may vary. Data for the specific rate coefficients for abstraction from CH bonds have been derived from experiments with hydrocarbons with different distributions of primary, secondary, and tertiary CH bonds. A primary CH bond is one on a carbon that is only connected to one other carbon, that is, the end carbon in a chain or a branch of a chain of carbon atoms. A secondary CH bond is one on a carbon atom connected to two others, and a tertiary CH bond is on a carbon atom that is

121

Explosive and General Oxidative Characteristics of Fuels

connected to three others. In a chain the CH bond strength on the carbons second from the ends is a few kilojoules less than other secondary atoms. The tertiary CH bond strength is still less, and the primary is the greatest. Assuming additivity of these rates, one can derive specific reaction rate constants for abstraction from the higher-order hydrocarbons by H, O, OH, and HO2 [31]. From the rates given in Ref. [31], the relative magnitudes of rate constants for abstraction of H by H, O, OH, and HO2 species from single tertiary, secondary, and primary CH bonds at 1080 K have been determined [32]. These relative magnitudes, which should not vary substantially over modest ranges of temperatures, were found to be as listed here:

Tertiary

:

Secondary

:

Primary

H

13

:

4

:

1

O

10

:

5

:

1

OH

4

:

3

:

1

HO2

10

:

3

:

1

Note that the OH abstraction reaction, which is more exothermic than the others, is the least selective of H atom position in its attack on large hydrocarbon molecules. There is also great selectivity by H, O, and HO2 between tertiary and primary CH bonds. Furthermore, estimates of rate constants at 1080 K [31] and radical concentrations for a reacting hydrocarbon system [33] reveal that the k values for H, O, and OH are practically the same and that during early reaction stages, when concentrations of fuel are large, the radical species concentrations are of the same order of magnitude. Only the HO2 rate constant departs from this pattern, being lower than the other three. Consequently, if one knows the structure of a paraffin hydrocarbon, one can make estimates of the proportions of various radicals that would form from a given fuel molecule [from the abstraction reaction (3.98)]. The radicals then decay further according to R(M) → olefin  R(M)

(3.99)

where R is a H atom or another hydrocarbon radical. The ethyl radical will thus become ethene and a H atom. Propane leads to an n-propyl and isopropyl radical:

H

H

H

H

C

C

C

H

H (Isopropyl)

H

H

H

H

H

C

C

C

H

H

H

(n-Propyl)

122

Combustion

These radicals decompose according to the β-scission rule, which implies that the bond that will break is one position removed from the radical site, so that an olefin can form without a hydrogen shift. Thus the isopropyl radical gives propene and a H atom, while the n-propyl radical gives ethene and a methyl radical. The β-scission rule states that when there is a choice between a CC single bond and a CH bond, the CC bond is normally the one that breaks because it is weaker than the CH bond. Even though there are six primary CH bonds in propane and these are somewhat more tightly bound than the two secondary ones, one finds substantially more ethene than propene as an intermediate in the oxidation process. The experimental results [12] shown in Fig. 3.12 verify this conclusion. The same experimental effort found the olefin trends shown in Table 3.2. Note that it is possible to estimate the order reported from the principles just described. If the initial intermediate or the original fuel is a large monoolefin, the radicals will abstract H from those carbon atoms that are singly bonded because the CH bond strengths of doubly bonded carbons are large (see Appendix D). Thus, the evidence [12, 32] is building that, during oxidation, all nonaromatic hydrocarbons primarily form ethene and propene (and some butene and isobutene) and that the oxidative attack that eventually leads to CO is almost solely from these small intermediates. Thus the study of ethene oxidation is crucially important for all alkyl hydrocarbons. It is also necessary to explain why there are parentheses around the collision partner M in reactions (3.94), (3.95), and (3.99). When RH in reactions (3.94) and (3.95) is ethane and R in reaction (3.99) is the ethyl radical, the reaction order depends on the temperature and pressure range. Reactions (3.94), (3.95), and (3.99) for the ethane system are in the fall-off regime for most typical combustion conditions. Reactions (3.94) and (3.95) for propane may lie in the fall-off regime for some combustion conditions; however, around 1 atm, butane and larger molecules pyrolyze near their high-pressure limits [34] and essentially follow first-order kinetics. Furthermore, for the formation of the olefin, an ethyl radical in reaction (3.99) must compete with the abstraction reaction. C2 H 5  O2 → C2 H 4  HO2

(3.100)

Owing to the great instability of the radicals formed from propane and larger molecules, reaction (3.99) is fast and effectively first-order; thus, competitive reactions similar to (3.100) need not be considered. Thus, in reactions (3.94) and (3.95) the M has to be included only for ethane and, to a small degree, propane; and in reaction (3.99) M is required only for ethane. Consequently, ethane is unique among all paraffin hydrocarbons in its combustion characteristics. For experimental purposes, then, ethane (like methane) should not be chosen as a typical hydrocarbon fuel.

123

Explosive and General Oxidative Characteristics of Fuels

c. Olefin and Acetylene Oxidation Following the discussion from the preceding section, consideration will be given to the oxidation of ethene and propene (when a radical pool already exists) and, since acetylene is a product of this oxidation process, to acetylene as well. These small olefins and acetylene form in the oxidation of a paraffin or any large olefin. Thus, the detailed oxidation mechanisms for ethane, propane, and other paraffins necessarily include the oxidation steps for the olefins [28]. The primary attack on ethene is by addition of the biradical O, although abstraction by H and OH can play some small role. In adding to ethene, O forms an adduct [35] that fragments according to the scheme

H C H

O

C H

#

H

H H

C

C

H

O

H

CH3  HCO

CH2  H2CO

The primary products are methyl and formyl radicals [36, 37] because potential energy surface crossing leads to a H shift at combustion temperatures [35]. It is rather interesting that the decomposition of cyclic ethylene oxide proceeds through a route in which it isomerizes to acetaldehyde and readily dissociates into CH3 and HCO. Thus two primary addition reactions that can be written are C2 H 4  O → CH3  HCO

(3.101)

C2 H 4  O → CH 2  H 2 CO

(3.102)

Another reaction—the formation of an adduct with OH—has also been suggested [38]: C2 H 4  OH → CH3  H 2 CO

(3.103)

However, this reaction has been questioned [4] because it is highly endothermic. OH abstraction via C2 H 4  OH → C2 H3  H 2 O

(3.104)

124

Combustion

could have a rate comparable to the preceding three addition reactions, and H abstraction C2 H 4  H → C2 H 3  H 2

(3.105)

could also play a minor role. Addition reactions generally have smaller activation energies than abstraction reactions; so at low temperatures the abstraction reaction is negligibly slow, but at high temperatures the abstraction reaction can dominate. Hence the temperature dependence of the net rate of disappearance of reactants can be quite complex. The vinyl radical (C2H3) decays to acetylene primarily by C2 H 3  M → C2 H 2  H  M

(3.106)

but, again, under particular conditions the abstraction reaction C2 H3  O2 → C2 H 2  HO2

(3.107)

must be included. Other minor steps are given in Appendix C. Since the oxidation mechanisms of CH3, H2CO (formaldehyde), and CO have been discussed, only the fate of C2H2 and CH2 (methylene) remains to be determined. The most important means of consuming acetylene for lean, stoichiometric, and even slightly rich conditions is again by reaction with the biradical O [37, 39, 39a] to form a methylene radical and CO, C2 H 2  O → CH 2  CO

(3.108)

through an adduct arrangement as described for ethene oxidation. The rate constant for reaction (3.108) would not be considered large in comparison with that for reaction of O with either an olefin or a paraffin. Mechanistically, reaction (3.108) is of significance. Since the C2H2 reaction with H atoms is slower than H  O2, the oxidation of acetylene does not significantly inhibit the radical pool formation. Also, since its rate with OH is comparable to that of CO with OH, C2H2—unlike the other fuels discussed—will not inhibit CO oxidation. Therefore substantial amounts of C2H2 can be found in the high-temperature regimes of flames. Reaction (3.108) states that acetylene consumption depends on events that control the O atom concentration. As discussed in Chapter 8, this fact has implications for acetylene as the soot-growth species in premixed flames. Acetylene–air flame speeds and detonation velocities are fast primarily because high temperatures evolve, not necessarily because acetylene reaction

125

Explosive and General Oxidative Characteristics of Fuels

mechanisms contain steps with favorable rate constants. The primary candidate to oxidize the methylene formed is O2 via CH 2  O2 → H 2 CO  O

(3.109)

however, some uncertainty attaches to the products as specified. Numerous other possible reactions can be included in a very complete mechanism of any of the oxidation schemes of any of the hydrocarbons discussed. Indeed, the very fact that hydrocarbon radicals form is evidence that higher-order hydrocarbon species can develop during an oxidation process. All these reactions play a very minor, albeit occasionally interesting, role; however, their inclusion here would detract from the major steps and important insights necessary for understanding the process. With respect to propene, it has been suggested [35] that O atom addition is the dominant decay route through an intermediate complex in the following manner: H

H C CH3

CH2  O

O CH2

C CH3

C2H5

O

C H

M

C2H5  HCO

X

H5C2CO  XH

For the large activated propionaldehyde molecule, the pyrolysis step appears to be favored and the equilibrium with the propylene oxide shifts in its direction. The products given for this scheme appear to be consistent with experimental results [38]. The further reaction history of the products has already been discussed. Essentially, the oxidation chemistry of the aliphatics higher than C2 has already been discussed since the initiation step is mainly CC bond cleavage with some CH bond cleavage. But the initiation steps for pure ethene or acetylene oxidation are somewhat different. For ethene the major initiation steps are [4, 39a] C2 H 4  M → C2 H 2  H 2  M

(3.110)

C2 H 4  M → C2 H 3  H  M

(3.111)

Reaction (3.110) is the fastest, but reaction (3.111) would start the chain. Similarly, the acetylene initiation steps [4] are C2 H 2  M → C2 H  H  M

(3.112)

C2 H 2  C2 H 2 → C 4 H 3  H  M

(3.113)

126

Combustion

Reaction (3.112) dominates under dilute conditions and reaction (3.113) is more important at high fuel concentrations [4]. The subsequent history of C2H and C4H3 is not important for the oxidation scheme once the chain system develops. Nevertheless, the oxidation of C2H could lead to chemiluminescent reactions that form CH and C2, the species responsible for the blue-green appearance of hydrocarbon flames. These species may be formed by the following steps [40, 40a, 40b]: C2 H  O → CH*  CO C2 H  O2 → CH*  CO2 CH  H → C  H 2 C  CH → C*2  H CH  CH → C*2  H 2 where the asterisk (*) represents electronically excited species. Taking all these considerations into account, it is possible to postulate a general mechanism for the oxidation of aliphatic hydrocarbons; namely, ⎧⎪ R I  R II (M)⎫⎪ ⎪⎧⎪ M ⎪⎫⎪ ⎪⎪ ⎪⎪ ⎪ ⎪ RH  ⎨ M ⎬ → ⎪⎨ R I  H(M) ⎪⎬ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪⎩ O2 ⎪⎪⎭ ⎪⎪⎩ R I  HO2 ⎪⎪⎭ where the H creates the radical pool (X  H, O, and OH) and the following occurrences: RH  X → R  XH   ethyl only) ⎧⎪ HO2 (R ⎪⎪ ⎧⎪⎪ O2 ⎫⎪⎪ ⎪   ⎨ ⎬ → Olefin  ⎨ H  M (R   ethyl and propyl only) R ⎪⎪⎩ M ⎪⎪⎭ ⎪⎪  III (M) ⎪⎪⎩ R ↓   CH ) Olefin (except for R 3 ⎛ ethene ⎞⎟ ⎪⎧⎪ O ⎪⎫⎪ ⎜ ⎛ ⎞⎟ ⎟ ⎪ ⎪  IV  ⎜⎜ formyl or acetyl radical, Olefin ⎜⎜⎜ and ⎟⎟  ⎨ H ⎬ → R ⎟ ⎜ ⎟ ⎪ ⎪ ⎝ formaldehyde, acetylene, or CO ⎟⎟⎠ ⎜⎜ propene ⎟ ⎪ ⎝ ⎠ ⎪⎪⎩ OH ⎪⎪⎪⎭ C2 H 2  O → CH 2  CO CO  OH → CO2  H

Explosive and General Oxidative Characteristics of Fuels

127

As a matter of interest, the oxidation of the diolefin butadiene appears to occur through O atom addition to a double bond as well as through abstraction reactions involving OH and H. Oxygen addition leads to 3-butenal and finally alkyl radicals and CO. The alkyl radical is oxidized by O atoms through acrolein to form CO, acetylene, and ethene. The abstraction reactions lead to a butadienyl radical and then vinyl acetylene. The butadienyl radical is now thought to be important in aromatic ring formation processes in soot generation [41–43]. Details of butadiene oxidation are presented in Ref. [44].

2. Alcohols Consideration of the oxidation of alcohol fuels follows almost directly from Refs. [45, 46]. The presence of the OH group in alcohols makes alcohol combustion chemistry an interesting variation of the analogous paraffin hydrocarbon. Two fundamental pathways can exist in the initial attack on alcohols. In one, the OH group can be displaced while an alkyl radical also remains as a product. In the other, the alcohol is attacked at a different site and forms an intermediate oxygenated species, typically an aldehyde. The dominant pathway depends on the bond strengths in the particular alcohol molecule and on the overall stoichiometry that determines the relative abundance of the reactive radicals. For methanol, the alternative initiating mechanisms are well established [47–50]. The dominant initiation step is the high-activation process CH3 OH  M → CH3  OH  M

(3.114)

which contributes little to the products in the intermediate (1000 K) temperature range [49]. By means of deuterium labeling, Aders [51] has demonstrated the occurrence of OH displacement by H atoms: CH3 OH  H → CH3  H 2 O

(3.115)

This reaction may account for as much as 20% of the methanol disappearance under fuel-rich conditions [49]. The chain branching system originates from the reactions CH3 OH  M → CH 2 OH  H

and

CH3 OH  H → CH 2 OH  H 2

which together are sufficient, with reaction (3.117) below, to provide the chain. As in many hydrocarbon processes, the major oxidation route is by radical abstraction. In the case of methanol, this yields the hydroxymethyl radical and, ultimately, formaldehyde via CH3 OH  X → CH 2 OH  XH

(3.116)

128

Combustion

⎪⎧ M ⎪⎫ ⎪⎧ H  M ⎪⎫ CH 2 OH  ⎨ ⎬ → H 2 CO  ⎨ ⎬ ⎪⎪⎩ O2 ⎪⎪⎭ ⎪⎪⎩ HO2 ⎪⎪⎭

(3.117)

where as before X represents the radicals in the system. Radical attack on CH2OH is slow because the concentrations of both radicals are small owing to the rapid rate of reaction (3.116). Reactions of OH and H with CH3OH to form CH3O (vs. CH2OH) and H2O and H2, respectively, have also been found to contribute to the consumption of methanol [49a]. These radical steps are given in Appendix C. The mechanism of ethanol oxidation is less well established, but it apparently involves two mechanistic pathways of approximately equal importance that lead to acetaldehyde and ethene as major intermediate species. Although in flow-reactor studies [45] acetaldehyde appears earlier in the reaction than does ethene, both species are assumed to form directly from ethanol. Studies of acetaldehyde oxidation [52] do not indicate any direct mechanism for the formation of ethene from acetaldehyde. Because C¶C bonds are weaker than the C¶OH bond, ethanol, unlike methanol, does not lose the OH group in an initiation step. The dominant initial step is C2 H 5 OH  M → CH3  CH 2 OH  M

(3.118)

As in all long-chain fuel processes, this initiation step does not appear to contribute significantly to the product distribution and, indeed, no formaldehyde is observed experimentally as a reaction intermediate. It appears that the reaction sequence leading to acetaldehyde would be C2 H 5 OH  X → CH3 CHOH  XH

(3.119)

⎪⎧ M ⎪⎫ ⎪⎧ H  M ⎪⎫ CH3 CHOH  ⎨ ⎬ → CH3 CHO  ⎨ ⎬ ⎪⎪⎩ O2 ⎪⎪⎭ ⎪⎪⎩ HO2 ⎪⎪⎭

(3.120)

By analogy with methanol, the major source of ethene may be the displacement reaction C2 H 5 OH  H → C2 H 5  H 2 O

(3.121)

with the ethyl radical decaying into ethene. Because the initial oxygen concentration determines the relative abundance of specific abstracting radicals, ethanol oxidation, like methanol oxidation, shows a variation in the relative concentration of intermediate species according to the overall stoichiometry. The ratio of acetaldehyde to ethene increases for lean mixtures.

Explosive and General Oxidative Characteristics of Fuels

129

As the chain length of the primary alcohols increases, thermal decomposition through fracture of C¶C bonds becomes more prevalent. In the pyrolysis of n-butanol, following the rupture of the C3H7¶CH2OH bond, the species found are primarily formaldehyde and small hydrocarbons. However, because of the relative weakness of the C¶OH bond at a tertiary site, t-butyl alcohol loses its OH group quite readily. In fact, the reaction t ---C4 H 9 OH → i ---C4 H8  H 2 O

(3.122)

serves as a classic example of unimolecular thermal decomposition. In the oxidation of t-butanol, acetone and isobutene appear [46] as intermediate species. Acetone can arise from two possible sequences. In one, (CH3 )3 COH → (CH3 )2 COH  CH3

(3.123)

(CH3 )2 COH  X → CH3 COCH3  XH

(3.124)

and in the other, H abstraction leads to β-scission and a H shift as C4 H 9 OH  X → C4 H8 OH  XH

(3.125)

C4 H8 OH → CH3 COCH3  CH3

(3.126)

Reaction (3.123) may be fast enough at temperatures above 1000 K to be competitive with reaction (3.122) [53].

3. Aromatic Hydrocarbons As discussed by Brezinsky [54], the oxidation of benzene and alkylated aromatics poses a problem different from the oxidation of aliphatic fuels. The aromatic ring provides a site for electrophilic addition reactions that effectively compete with the abstraction of H from the ring itself or from the side chain. When the abstraction reactions involve the side chain, the aromatic ring can strongly influence the degree of selectivity of attack on the side chain hydrogens. At high enough temperatures the aromatic ring thermally decomposes and thereby changes the whole nature of the set of hydrocarbon species to be oxidized. As will be discussed in Chapter 4, in flames the attack on the fuel begins at temperatures below those where pyrolysis of the ring would be significant. As the following sections will show, the oxidation of benzene can follow a significantly different path than that of toluene and other higher alkylated aromatics. In the case of toluene, its oxidation bears a resemblance to that of methane; thus it, too, is different from benzene and other alkylated aromatics.

130

Combustion

In order to establish certain terms used in defining aromatic reactions, consider the following, where the structure of benzene is represented by the symbol . Abstraction reaction: H

H H

H

C

C C

C HX

H

H H

H

H H

H

C

C C

C H  XH

H

H H

H

Displacement reaction: H

H H

H

C

C C

C HH

H

H H

H



H

H H

H

C

C C

C

H

H H

H

H

Homolysis reaction: H

H H

H

H

C

C C

C H

C

H

H H

H

H



H

H H

C

C C

H

H H

H

Addition reaction: H O

OH

 O

a. Benzene Oxidation Based on the early work of Norris and Taylor [55] and Bernard and lbberson [56], who confirmed the theory of multiple hydroxylation, a general low-temperature oxidation scheme was proposed [57, 58]; namely, C6 H6  O2 → C6 H 5  HO2

(3.127)

C6 H 5  O2  C6 H 5 OO

(3.128)

131

Explosive and General Oxidative Characteristics of Fuels

C6 H 5 OO  C6 H6 → C6 H 5 OOH  C6 H 5 ↓ C6 H 5 O  OH

(3.129)

C6 H 5 O  C6 H6 → C6 H 5 OH  C6 H 5

(3.130)

sequence C6 H 5 OH  O2 ⎯ ⎯⎯⎯⎯ → C6 H 4 (OH)2 as above

(3.131)

There are two dihydroxy benzenes that can result from reaction (3.131)— hydroquinone and pyrocatechol. It has been suggested that they react with oxygen in the following manner [55]: OH OH

OH



C

O

HC

O

HC

O



O2

OH

C

OH

C2H2

(3.132)

O OH

Hydroquinone

Triplet

OH  OH Pyrocatechol

Maleic acid

O

OH

O

O

C

OH

O

C

OH

2C2H2 

O2 OH

Triplet

Oxalic acid

(3.133) Thus maleic acid forms from the hydroquinone and oxalic acid forms from pyrocatechol. However, the intermediate compounds are triplets, so the intermediate steps are “spin-resistant” and may not proceed in the manner indicated. The intermediate maleic acid and oxalic acid are experimentally detected in this low-temperature oxidation process. Although many of the intermediates were detected in low-temperature oxidation studies, Benson [59] determined that the ceiling temperature for bridging peroxide molecules formed from aromatics was of the order of 300°C; that is, the reverse of reaction (3.128) was favored at higher temperatures. It is interesting to note that maleic acid dissociates to two carboxyl radicals and acetylene  ¨ O)  C H Maleic acid → 2(HO ¶ C 2 2

(3.134)

132

Combustion

while oxalic acid dissociates into two carboxyl radicals  ¨ O) Oxalic acid → 2(HO¶C

(3.135)

Under this low-temperature condition the carboxylic radical undergoes attack ⎧⎪ M ⎫⎪ ⎧⎪ H  M ⎫⎪ ⎪⎪ ⎪⎪ ⎪ ⎪  HO¶C ¨ O  ⎨ O2 ⎬ → CO2  ⎪⎨ HO2 ⎪⎬ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ ⎪⎩ X ⎪⎭ ⎪⎩ XH ⎪⎪⎭

(3.136)

to produce CO2 directly rather than through the route of CO oxidation by OH characteristic of the high-temperature oxidation of hydrocarbons. High-temperature flow-reactor studies [60, 61] on benzene oxidation revealed a sequence of intermediates that followed the order: phenol, cyclopentadiene, vinyl acetylene, butadiene, ethene, and acetylene. Since the sampling techniques used in these experiments could not distinguish unstable species, the intermediates could have been radicals that reacted to form a stable compound, most likely by hydrogen addition in the sampling probe. The relative time order of the maximum concentrations, while not the only criterion for establishing a mechanism, has been helpful in the modeling of many oxidation systems [4, 13]. As stated earlier, the benzene molecule is stabilized by strong resonance; consequently, removal of a H from the ring by pyrolysis or O2 abstraction is difficult and hence slow. It is not surprising, then, that the induction period for benzene oxidation is longer than that for alkylated aromatics. The high-temperature initiation step is similar to that of all the cases described before, that is,



M O2



HM HO2

(3.137)

Phenyl

but it probably plays a small role once the radical pool builds from the H obtained. Subsequent formation of the phenyl radical arises from the propagating step



O OH H



OH H2O H2

(3.138)

133

Explosive and General Oxidative Characteristics of Fuels

The O atom could venture through a displacement and possibly an addition [60] reaction to form a phenoxyl radical and phenol according to the steps

O

 O

 H

Phenoxy

(3.139)

OH

 O

Phenol

Phenyl radical reactions with O2, O, or HO2 seem to be the most likely candidates for the first steps in the aromatic ring-breaking sequence [54, 61]. A surprising metathesis reaction that is driven by the resonance stability of the phenoxy product has been suggested from flow-reactor studies [54] as a key step in the oxidation of the phenyl radical: O

 O2

 O

(3.140)

In comparison, the analogous reaction written for the methyl radical is highly endothermic. This chain branching step was found [61, 62] to be exothermic to the extent of approximately 46 kJ/mol, to have a low activation energy, and to be relatively fast. Correspondingly, the main chain branching step [reaction (3.17)] in the H2¶O2 system is endothermic to about 65 kJ/mol. This rapid reaction (3.138) would appear to explain the large amount of phenol found in flow-reactor studies. In studies [63] of near-sooting benzene flames, the low mole fraction of phenyl found could have required an unreasonably high rate of reaction (3.138). The difference could be due to the higher temperatures, and hence the large O atom concentrations, in the flame studies.

134

Combustion

The cyclopentadienyl radical could form from the phenoxy radical by O

O

CO 

(3.141)

Cyclopentyl dienyl

The expulsion of CO from ketocyclohexadienyl radical is also reasonable, not only in view of the data of flow-reactor results, but also in view of other pyrolysis studies [64]. The expulsion indicates the early formation of CO in aromatic oxidation, whereas in aliphatic oxidation CO does not form until later in the reaction after the small olefins form (see Figs. 3.11 and 3.12). Since resonance makes the cyclopentadienyl radical very stable, its reaction with an O2 molecule has a large endothermicity. One feasible step is reaction with O atoms; namely, O

H

O

O

(3.142) CO  H2C

H C

C H

CH

O

The butadienyl radical found in reaction (3.142) then decays along various paths [44], but most likely follows path (c) of reaction (3.143): H2 C

CH

C

CH  H

(a)

CH2  R

(b)

Vinyl acetylene

H2C

CH

CH

RH H2C

CH

CH

CH

Butadiene

H2C

CH  HC

CH

(c)

RH H2C

CH2  R (d)

HC

CH  H (e)

(3.143)

135

Explosive and General Oxidative Characteristics of Fuels

Although no reported work is available on vinyl acetylene oxidation, oxidation by O would probably lead primarily to the formation of CO, H2, and acetylene (via an intermediate methyl acetylene) [37]. The oxidation of vinyl acetylene, or the cyclopentadienyl radical shown earlier, requires the formation of an adduct [as shown in reaction (3.142)]. When OH forms the adduct, the reaction is so exothermic that it drives the system back to the initial reacting species. Thus, O atoms become the primary oxidizing species in the reaction steps. This factor may explain why the fuel decay and intermediate species formed in rich and lean oxidation experiments follow the same trend, although rich experiments show much slower rates [65] because the concentrations of oxygen atoms are lower. Figure 3.13 is a summary of the reaction steps that form the general mechanism of benzene and the phenyl radical oxidation based on a modified version of a model proposed by Emdee et al. [61, 66]. Other models of benzene oxidation [67, 68, which are based on Ref. [61], place emphasis on different reactions.

b. Oxidation of Alkylated Aromatics The initiation step in the high-temperature oxidation of toluene is the pyrolytic cleavage of a hydrogen atom from the methyl side chain, and at lower temperatures it is O2 abstraction of an H from the side chain, namely CH2

CH3



M O2

Toluene



HM HO2

(3.144)

Benzyl

The H2¶O2 radical pool that then develops begins the reactions that cause the fuel concentration to decay. The most effective attackers of the methyl side chain of toluene are OH and H. OH does not add to the ring, but rather abstracts a H from the methyl side chain. This side-chain H is called a benzylic H. The attacking H has been found not only to abstract the benzylic H, but also to displace the methyl group to form benzene and a methyl radical [69]. The reactions are then CH2

CH3

 X

 XH

CH3

(3.145)

 H

 CH3

136

Combustion

OH H

C6H5

O H O2

C5H6

OH H C2H3

O OH H

C5H5

OH

H

CO C H O 5 5

C4H5

C6H5OH

C5H4OH

HO2

O

C2H3 C2H2

C6H5O CO

C6H6

C2H3

O2

C5H4O

CO 2C2H2 FIGURE 3.13 Molar rates of progress for benzene oxidation in an atmospheric turbulent flow reactor. The thickness of the lines represents the relative magnitudes of certain species as they pass through each reaction pathway.

The early appearance of noticeable dibenzyl quantities in flow-reactor studies certainly indicates that significant paths to benzyl exist and that benzyl is a stable radical intermediate. The primary product of benzyl radical decay appears to be benzaldehyde [33, 61]:

H CH2

O

 O2

O 

C

O

H C

H

(3.146)

H 

Benzaldehyde

The reaction of benzyl radicals with O2 through an intermediate adduct may not be possible, as was found for reaction of methyl radical and O2 (indeed,

137

Explosive and General Oxidative Characteristics of Fuels

one may think of benzyl as a methyl radical with one H replaced by a phenyl group). However, it is to be noted that the reaction H CH2

O

C

H

O C

H

 O

H 

(3.147)

Benzaldehyde

has been shown [33] to be orders of magnitude faster than reaction (3.146). The fate of benzaldehyde is the same as that of any aldehyde in an oxidizing system, as shown by the following reactions that lead to phenyl radicals and CO: O

H C

Pyrolysis

 H

C

(3.148)

O

H  CO

O

O

H C

C

 X

 XH

(3.149)

 CO

Reaction (3.149) is considered as the major channel. Reaction (3.147) is the dominant means of oxidizing benzyl radicals. It is a slow step, so the oxidation of toluene is overall slower than that of benzene,

138

Combustion

even though the induction period for toluene is shorter. The oxidation of the phenyl radical has been discussed, so one can complete the mechanism of the oxidation of toluene by referring to that section. Figure 3.14 from Ref. [66] is an appropriate summary of the reactions. The first step of other high-order alkylated aromatics proceeds through pyrolytic cleavage of a CC bond. The radicals formed soon decay to give H atoms that initiate the H2¶O2 radical pool. The decay of the initial fuel is dominated by radical attack by OH and H, or possibly O and HO2, which abstract an H from the side chain. The benzylic H atoms (those attached to the carbon next to the ring) are somewhat easier to remove because of their lower O2

C6H5CH3

H OH H

HO2 C6H5CH2

O

H OH

CO

C2H3

O2 C6H5

H O

H O2

C5H6

OH H C2H3

O OH H

C5H5

CO C H O 5 5

OH

C6H5OH

C5H4OH H

C4H5

C6H5O

HO2

O

C2H3 C2H2

C6H5CO

H

CO

C6H6

O

OH C6H5CHO

C5H4O

CO 2C2H2 Benzene submechanism

FIGURE 3.14 Molar rates of progress for toluene oxidation in an atmospheric turbulent flow reactor (cf. to Fig 3.13). The benzene submechanism is outlined for toluene oxidation. Dashed arrows represent paths that are important to benzene oxidation, but not significant for toluene (from Ref. [66]).

139

Explosive and General Oxidative Characteristics of Fuels

bond strength. To some degree, the benzylic H atoms resemble tertiary or even aldehydic H atoms. As in the case of abstraction from these two latter sites, the case of abstraction of a single benzylic H can be quickly overwhelmed by the cumulative effect of a greater number of primary and secondary H atoms. The abstraction of a benzylic H creates a radical such as H

H

C

C

R

H

which by the β-scission rule decays to styrene and a radical [65] H

H

C

C

H H R

C

 R

C

(3.150)

H H

where R can, of course, be H if the initial aromatic is ethyl benzene. It is interesting that in the case of ethyl benzene, abstraction of a primary H could also lead to styrene. Apparently, two approximately equally important processes occur during the oxidation of styrene. One is oxidative attack on the doublebonded side chain, most probably through O atom attacks in much the same manner that ethylene is oxidized. This direct oxidation of the vinyl side chain of styrene leads to a benzyl radical and probably a formyl radical. The other is the side chain displacement by H to form benzene and a vinyl radical. Indeed the displacement of the ethyl side chain by the same process has been found to be a major decomposition route for the parent fuel molecule. If the side chain is in an “iso” form, a more complex aromatic olefin forms. Isopropyl benzene leads to a methyl styrene and styrene [70]. The long-chain alkylate aromatics decay to styrene, phenyl, benzyl, benzene, and alkyl fragments. The oxidation processes of the xylenes follow somewhat similar mechanisms [71, 72].

4. Supercritical Effects The subject of chemical reactions under supercritical conditions is well outside the scope of matters of major concern to combustion related considerations. However, a trend to increase the compression ratio of some turbojet engines has raised concerns that the fuel injection line to the combustion chamber could place the fuel in a supercritical state; that is; the pyrolysis of the fuel in the line could increase the possibility of carbon formations such as soot. The

140

Combustion

question then arises as to whether the pyrolysis of the fuel in the line could lead to the formation of PAH (polynuclear aromatic hydrocarbons) which generally arise in the chemistry of soot formation (see Chapter 8; Section E6). Since the general conditions in devises of concern are not near the critical point, then what is important in something like a hydrocarbon decomposition process is whether the high density of the fuel constituents affects the decomposition kinetic process so that species would appear other than those that would occur in a subcritical atmosphere. What follows is an attempt to give some insight into a problem that could arise in some cases related to combustion kinetics, but not necessarily related to the complete field of supercritical use as described in pure chemistry texts and papers. It is apparent that the high pressure in the supercritical regime not only affects the density (concentration) of the reactions, but also the diffusivity of the species that form during pyrolysis of important intermediates that occur in fuel pyrolysis. Indeed, as well, in considering the supercritical regime one must also be concerned that the normal state equation may not hold. In some early work on the pyrolysis of the endothermic fuel methylcyclohexane (MCH) it was found that in the subcritical state MCH pyrolysis is β scission dominated (see Chapter 3, Section H1) and little, if any, PAH are found [73, 74]. Also it was found that while β scission processes are still important under supercritical conditions, they are significantly slower [75]. These studies suggest that the pyrolysis reaction of MCH proceeds to form the methylhexedienly radical (MHL) under both sub- and supercritical conditions [73, 74]. However, under the supercritical condition it was found that dimethylcyclopentane subsequently forms. The process by which the initial 6-member ring is converted to a 5-member ring is most apparently due to the phenomenon of caging, a phenomenon frequently discussed in the supercritical chemical process literature [75]. The formation of a cyclic intermediate is more likely to produce PAH. Thus, once MHL forms it can follow two possible routes; β scission leading to ordinary pyrolysis or a cyclization due to the phenomenon called caging. The extent of either depends on the physical parameters of the experiment, essentially the density (or pressure). In order to estimate the effect of caging with respect to a chemical reaction process, the general approach has been to apply transition state theory [75]. What has been considered in general transition state theory (see Chapter 2, Section B2) is the rate of formation of a product through an intermediate (complex) in competition with the intermediate reforming the initial reactant. Thus β scission is considered in competition with caging. However, in essence, the preceding paragraph extends the transition state concept in that the intermediate does not proceed back to the reactant, but has two possible routes to form different products. One route is a β scission route to produce a general hydrocarbon pyrolysis product and the other is a caging process possibly leading to a product that can cause fuel line fouling. Following the general chemical approach [75] to evaluate the extent of a given route it is possible to conclude that under supercritical conditions the extent of fuel fouling (PAH formation) could be determined by the ratio of the

141

Explosive and General Oxidative Characteristics of Fuels

collision rate of formation of the new cyclohydrocarbon due to caging to the diffusion rate of the β scission products “to get out of the cage”. This ratio can be represented by the expression [νd2 exp(E/RT)/D] or [ν exp(E/RT)/(D/d2)], where ν is the collision frequency (s1), d2 the collision cross-section, E the activation energy, and D the mass diffusivity (cm2/s) [75]. The second ratio expression is formulated so that a ratio of characteristic times is presented. This time ratio will be recognized as a Damkohler number [75]. For the pyrolysis process referred to, the caging institutes a bond formation process and thus activation energy does not exist. Then the relevant Damkohler number is [ν/(D/d2)]. Typical small molecule diffusivities have been reported to be from 101 cm2/s for gases to 105 cm2/s for liquids [76]. One would estimate that under supercritical conditions that the supercritical fluid would be somewhere between the two values. It has been proposed that although supercritical fluids have in many instances greater similarity to liquids than gases, their diffusivities are more like gases than liquids. Thus, the caging product should increase with pressure, as has been found in MCH pyrolysis [74] and, perhaps, in other similar cases related to combustion problems.

PROBLEMS (Those with an asterisk require a numerical solution and use of an appropriate software program—See Appendix I.) 1. In hydrocarbon oxidation a negative reaction rate coefficient is possible. (a) What does this statement mean and when does the negative rate occur? (b) What is the dominant chain branching step in the high-temperature oxidation of hydrocarbons? (c) What are the four dominant overall steps in the oxidative conversion of aliphatic hydrocarbons to fuel products? 2. Explain in a concise manner what the essential differences in the oxidative mechanisms of hydrocarbons are under the following conditions: a. The temperature is such that the reaction is taking place at a slow (measurable) rate—for example, a steady reaction. b. The temperature is such that the mixture has just entered the explosive regime. c. The temperature is very high, like that obtained in the latter part of a flame or in a shock tube. Assume that the pressure is the same in all three cases. 3. Draw the chemical structure of heptane, 3-octene, and isopropyl benzene. 4. What are the first two species to form during the thermal dissociation of each of the following radicals?

CH3 H3C

C

H3C CH3

H

H

H

C

C

C

H

H

CH3 H3C

H

CH3

C

C

C

H

H

H

CH3

142

Combustion

5. Toluene is easier to ignite than benzene, yet its overall burning rate is slower. Explain why. 6. Determine the generalized expression for the α criteria when the only termination step is radical-radical recombination. 7. In examining Equation 3.8, what is the significance of the condition? k2 (α  1)(M) k4 (M)  k5 (O2 )(M) 8. Tetra ethyl lead (TEL) was used as an anti-knock agent in automotive gasoline. Small amounts were normally added. During the compression stroke TEL reacts with the air to form very small lead oxide particles. Give an explanation why you believe TEL would be an effective anti-knock agent. 9.*The reaction rate of a dilute mixture of stoichiometric hydrogen and oxygen in N2 is to be examined at 950 K and 10 atm and compared to the rate at 950 K and 0.5 atm. The mixture consists of 1% (by volume) H2. Perform an adiabatic constant pressure calculation of the reaction kinetics at the two different conditions for a reaction time that allows you to observe the complete reaction. Use a chemical kinetics program such as SENKIN (CHEMKIN II and III) or the CLOSED HOMOGENEOUS_TRANSIENT code (CHEMKIN IV) to perform the calculation. Plot the temperature and major species profiles as a function of time. Discuss and explain the differences in the reaction rates. Use a sensitivity analysis or rate-of-progress analysis to assist your discussion. The reaction mechanism can be obtained from Appendix C or may be downloaded from the internet (for example, from the database of F. L. Dryer at http:// www.princeton.edu/~combust/database/other.html, the database from LLNL at http://www-cmls.llnl.gov/?url  science_and_technology-chemistry-combustion, or the database from Leeds University, http://www.chem.leeds.ac.uk/ Combustion/Combustion.html). 10.*Investigate the effect of moisture on the carbon monoxide–oxygen reaction by performing a numerical analysis of the time-dependent kinetics, for example, by using SENKIN (CHEMKIN II and III) or the CLOSED HOMOGENEOUS_TRANSIENT code (CHEMKIN IV). Assume a constant pressure reaction at atmospheric pressure, an initial temperature of 1150 K, and a reaction time of approximately 1 s. Choose a mixture consisting initially of 1% CO and 1% O2 with the balance N2 (by volume). Add to this mixture, various amounts of H2O starting from 0 ppm, 100 ppm, 1000 ppm, and 1% by volume. Plot the CO and temperature profiles for the different water concentrations and explain the trends. From the initial reaction rate, what is the overall reaction order with respect to water concentration. Use a mechanism from Appendix C or download one from the internet. 11.*Calculate the reaction kinetics of a methane oxygen mixture diluted with N2 at a constant pressure of 1 atm and initial temperature of 1100 K. Assume an adiabatic reaction with an initial concentration of CH4 of

Explosive and General Oxidative Characteristics of Fuels

143

1% by volume, an equivalence ratio of 0.2, and the balance of the mixture nitrogen. Use the GRI-Mech 3.0 chemical mechanism for methane oxidation (which may be obtained from G. P. Smith, D. M. Golden, M. Frenklach, N. W. Moriarty, B. Eiteneer, M. Goldenberg, C. T. Bowman, R. K. Hanson, S. Song, W. C. Gardiner, Jr., V. V. Lissianski, and Z. Qin, http://www.me.berkeley.edu/gri_mech/). Plot the major species profiles and temperature as a function of time. Determine the induction and ignition delay times of the mixture. Also, analyze the reaction pathways of methyl radicals with sensitivity and rate-of-progress analyses. 12.* Compare the effects of pressure on the reaction rate and mechanism of methane (see Problem 8) and methanol oxidation. Calculate the time-dependent kinetics for each fuel at pressures of 1 and 20 atm and an initial temperature of 1100 K. Assume the reaction occurs at constant pressure and the mixture consists of 1% by volume fuel, 10% by volume O2, and the balance of the mixture is nitrogen. A methanol mechanism may be obtained from the database of F. L. Dryer at http://www.princeton.edu/ ~combust/database/other.html. 13.*Natural gas is primarily composed of methane, with about 2–5% by volume ethane, and smaller concentrations of larger hydrocarbons. Determine the effect of small amounts of ethane on the methane kinetics of Problem 8 by adding to the fuel various amounts of ethane up to 5% by volume maintaining the same total volume fraction of fuel in the mixture. In particular, discuss and explain the effects of ethane on the induction and ignition delay periods.

REFERENCES 1. Dainton, F. S., “Chain Reactions: An Introduction,” 2nd Ed. Methuen, London, 1966. 2. Lewis, B., and von Elbe, G., “Combustion, Flames and Explosions of Gases,” 2nd Ed., Part 1 Academic Press, New York, 1961. 3. Gardiner, W. C. Jr., and Olson, D. B., Annu. Rev. Phys. Chem. 31, 377 (1980). 4. Westbrook, C. K., and Dryer, F. L., Prog. Energy Combust. Sci. 10, 1 (1984). 5. Bradley, J. N., “Flame and Combustion Phenomena,” Chap. 2. Methuen, London, 1969. 6. Baulch, D. L. et al., “Evaluated Kinetic Data for High Temperature Reactions,” Vol. 1–3. Butterworth, London, 1973. 6a. Yetter, R. A., Dryer, F. L., and Golden, D. M., in “Major Topics in Combustion.” (M. Y. Hussaini, A. Kumar, and R. G. Voigt, eds.), p. 309. Springer-Verlag, New York, 1992. 7. Westbrook, C. K., Combust. Sci. Technol. 29, 67 (1982). 7a. Michael, J. V., Sutherland, J. W., Harding, L. B., and Wagner, A. F., Proc. Combust. Inst. 28, 1471 (2000). 8. Brokaw, R. S., Proc. Combust. Inst. 11, 1063 (1967). 9. Gordon, A. S., and Knipe, R., J. Phys. Chem. S9, 1160 (1955). 10. Dryer, F. L., Naegeli, D. W., and Glassman, I., Combust. Flame 17, 270 (1971). 11. Smith, I. W. M., and Zellner, R., J.C.S. Faraday Trans. 269, 1617 (1973). 12. Larson, C. W., Stewart, P. 14, and Golden, D. M., Int. J. Chem. Kinet. 20, 27 (1988). 13. Dryer, F. L., and Glassman, I., Prog. Astronaut. Aeronaut. 62, 55 (1979). 14. Semenov, N. N., “Some Problems in Chemical Kinetics and Reactivity,” Chap. 7. Princeton University Press, Princeton, NJ, 1958.

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15. Minkoff, G. J., and Tipper, C. F. H., “Chemistry of Combustion Reactions.” Butterworth, London, 1962. 16. Williams, F. W., and Sheinson, R. S., Combust. Sci. Technol. 7, 85 (1973). 17. Bozzelli, J. W., and Dean, A. M., J. Phys. Chem. 94, 3313 (1990). 17a. Wagner, A. F., Slagle, I. R., Sarzynski, D., and Gutman, D., J. Phys. Chem. 94, 1853 (1990). 17b. Benson, S. W., Prog. Energy Combust. Sci. 7, 125 (1981). 18. Benson, S. W., NBS Spec. Publ. (US) 16(No. 359), 101 (1972). 19. Baldwin, A. C., and Golden, D. M., Chem. Phys. Lett. 55, 350 (1978). 20. Dagaut, P., and Cathonnet, M., J. Chem. Phys. 87, 221 (1990). 20a. Frenklach, M., Wang, H., and Rabinovitz, M. J., Prog. Energy Combust. Sci. 18, 47 (1992). 20b. Hunter, J. B., Wang, H., Litzinger, T. A., and Frenklach, M., Combust. Flame 97, 201 (1994). 21. GRI-Mech 2.11, available through the World Wide Web, http://www.me.berkeley. edu/gri_mech/ 22. Brabbs, T. A., and Brokaw, R. S., Proc. Combust. Inst. 15, 892 (1975). 23. Yu, C. -L., Wang, C., and Frenklach, M., J. Phys. Chem. 99, 1437 (1995). 24. Warnatz, J., Proc. Combust. Inst. 18, 369 (1981). 25. Warnatz, J., Prog. Astronaut. Aeronaut. 76, 501 (1981). 26. Dagaut, P., Cathonnet, M., and Boettner, J. C., Int. J. Chem. Kinet. 23, 437 (1991). 27. Dagaut, P., Cathonnet, M., Boettner, J. C., and Guillard, F., Combust. Sci. Technol. 56, 232 (1986). 28. Cathonnet, M., Combust. Sci. Technol. 98, 265 (1994). 29. Fristrom, R. M., and Westenberg, A. A., “Flame Structure,” Chap. 14. McGraw-Hill, New York, 1965. 30. Dryer, F. L., and Brezinsky, K., Combust. Sci. Technol. 45, 199 (1986). 31. Warnatz, J., in “Combustion Chemistry.” (W. C. Gardiner, Jr. ed.), Chap. 5. Springer-Verlag, New York, 1984. 32. Dryer, F. L., and K. Brezinsky, West. States Sect. Combust. Inst. Pap. No. 84–88 (1984). 33. Brezinsky, K., Litzinger, T. A., and Glassman, I., Int. J. Chem. Kinet. 16, 1053 (1984). 34. Golden, D. M., and Larson, C. W., Proc. Combust. Inst. 20, 595 (1985). 35. Hunziker, H. E., Kneppe, H., and Wendt, H. R., J. Photochem. 17, 377 (1981). 36. Peters, I., and Mahnen, G., in “Combustion Institute European Symposium.” (F. J. Weinberg, ed.), p. 53. Academic Press, New York, 1973. 37. Blumenberg, B., Hoyermann, K., and Sievert, R., Proc. Combust. Inst. 16, 841 (1977). 38. Tully, F. P., Phys. Chem. Lett. 96, 148 (1983), 19, 181 (1982). 39. Miller, J. A., Mitchell, R. E., Smooke, M. D., and Kee, R. J. Proc. Combust. Inst. 19, 181 (1982). 39a. Kiefer, J. H., Kopselis, S. A., Al-Alami, M. Z., and Budach, K. A., Combust. Flame 51, 79 (1983). 40. Grebe, J., and Homann, R. H., Ber. Busenges. Phys. Chem. 86, 587 (1982). 40a. Devriendt, K., and Peeters, J., J. Phys. Chem. A. 101, 2546 (1997). 40b. Marques, C. S. T., Benvenutti, L. H., and Betran, C. A., J. Braz. Chem. Soc. 17, 302 (2006). 41. Glassman, I. Mech. Aerosp. Eng., Rep. No. 1450. Princeton University, Princeton, NJ, 1979. 42. Cole, J. A., M.S. Thesis, Department of Chemical Engineering, MIT, Cambridge, MA, 1982. 43. Frenklach, M., Clary, D. W., Gardiner, W. C. Jr., and Stein, S. E., Proc. Combust. Inst. 20, 887 (1985). 44. Brezinsky, K., Burke, E. J., and Glassman, I., Proc. Combust. Inst. 20, 613 (1985). 45. Norton, T. S., and Dryer, F. L., Combust. Sci. Technol. 63, 107 (1989). 46. Held, T. J., and Dryer, F. L., Proc. Combust. Inst. 25, 901 (1994). 47. Aronowitz, D., Santoro, R. J., Dryer, F. L., and Glassman, I., Proc. Combust. Inst. 16, 663 (1978). 48. Bowman, C. T., Combust. Flame. 25, 343 (1975). 49. Westbrook, C. K., and Dryer, F. L., Combust. Sci. Technol. 20, 125 (1979). 49a. Li, J., Zhao, Z., Kazakov, A., Chaos, M., Dryer, F. L., and Scire, J. J. Jr., Int. J. Chem. Kinet. 39, 109 (2007).

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50. Vandooren, J., and van Tiggelen, P. J., Proc. Combust. Inst. 18, 473 (1981). 51. Aders, W. K., in “Combustion Institute European Symposium.” (F. J. Weinberg, ed.), p. 79. Academic Press, New York, 1973. 52. Colket, M. B. III, Naegeli, D. W., and Glassman, I., Proc. Combust. Inst. 16, 1023 (1977). 53. Tsang, W., J. Chem. Phys. 40, 1498 (1964). 54. Brezinsky, K., Prog. Energy Combust. Sci. 12, 1 (1986). 55. Norris, R. G. W., and Taylor, G. W., Proc. R. Soc. London, Ser. A. 153, 448 (1936). 56. Barnard, J. A., and lbberson, V. J., Combust. Flame 9(81), 149 (1965). 57. Glassman, I., Mech. Aerosp. Eng. Rep., No. 1446. Princeton University, Princeton, NJ, 1979. 58. Santoro, R. J., and Glassman, I., Combust. Sci. Technol. 19, 161 (1979). 59. Benson, S. W., J. Am Chem. Soc. 87, 972 (1965). 60. Lovell, A. B., Brezinsky, K., and Glassman, I., Proc. Combust. Inst. 22, 1065 (1985). 61. Emdee, J. L., Brezinsky, K., and Glassman, I., J. Phys. Chem. 96, 2151 (1992). 62. Frank, P., Herzler, J., Just, T., and Wahl, C., Proc. Combust. Inst. 25, 833 (1994). 63. Bittner, J. D., and Howard, J. B., Proc. Combust. Inst. 19, 221 (1977). 64. Lin, C. Y., and Lin, M. C., J. Phys. Chem. 90, 1125 (1986). 65. Litzinger, T. A., Brezinsky, K., and Glassman, I., Combust. Flame 63, 251 (1986). 66. Davis, S. C., “Personal communication.” Princeton University, Princeton, NJ, 1996. 67. Lindstedt, R. P., and Skevis, G., Combust. Flame 99, 551 (1994). 68. Zhang, H. -Y., and McKinnon, J. T., Combust. Sci. Technol. 107, 261 (1995). 69. Asthoiz, D. C., Durant, J., and Troe, J., Proc. Combust. Inst. 18, 885 (1981). 70. Litzinger, T. A., Brezinsky, K., and Glassman, I., J. Phys. Chem. 90, 508 (1986). 71. Emdee, J. L., Brezinsky, K., and Glassman, I., Proc. Combust. Inst. 23, 77 (1990). 72. Emdee, J. L., Brezinsky, K., and Glassman, I., J. Phys. Chem. 95, 1616 (1991). 73. Zeppieri, S., High Temperature Experimental and Computational Studies of the Pyrolysis and Oxidation of Endothermic Fuels, Ph.D. Thesis, Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, 1999. 74. Stewart, J., Supercritical Pyrolysis of the Endothermic Fuels Methylcyclohexane, Decalin, and Tetralin, Ph.D. Thesis, Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, 1999. 75. Wu, B. C., Klein, M. T., and Sandler, S. I., Ind. Eng. Chem. Res. 30, 822 (1991). 76. Ondruschka, A., Zimmermann, G., Remmler, M., Sedlackova, M., and Pola, A., J. Anal. Appl. Pyrol. 18, 19 (1990).

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Chapter 4

Flame Phenomena in Premixed Combustible Gases A. INTRODUCTION In the previous chapter, the conditions under which a fuel and oxidizer would undergo explosive reaction were discussed. Such conditions are strongly dependent on the pressure and temperature. Given a premixed fuel-oxidizer system at room temperature and ambient pressure, the mixture is essentially unreactive. However, if an ignition source applied locally raises the temperature substantially, or causes a high concentration of radicals to form, a region of explosive reaction can propagate through the gaseous mixture, provided that the composition of the mixture is within certain limits. These limits, called flammability limits, will be discussed in this chapter. Ignition phenomena will be covered in a later chapter. When a premixed gaseous fuel–oxidizer mixture within the flammability limits is contained in a long tube, a combustion wave will propagate down the tube if an ignition source is applied at one end. When the tube is opened at both ends, the velocity of the combustion wave falls in the range of 20–200 cm/s. For example, the combustion wave for most hydrocarbon–air mixtures has a velocity of about 40 cm/s. The velocity of this wave is controlled by transport processes, mainly simultaneous heat conduction and diffusion of radicals; thus it is not surprising to find that the velocities observed are much less than the speed of sound in the unburned gaseous mixture. In this propagating combustion wave, subsequent reaction, after the ignition source is removed, is induced in the layer of gas ahead of the flame front by two mechanisms that are closely analogous to the thermal and chain branching mechanisms discussed in the preceding chapter for static systems [1]. This combustion wave is normally referred to as a flame; and since it can be treated as a flow entity, it may also be called a deflagration. When the tube is closed at one end and ignited there, the propagating wave undergoes a transition from subsonic to supersonic speeds. The supersonic wave is called a detonation. In a detonation heat conduction and radical diffusion do not control the velocity; rather, the shock wave structure of the developed supersonic wave raises the temperature and pressure substantially to cause explosive reaction and the energy release that sustains the wave propagation. 147

148

Combustion

1

FIGURE 4.1

Unburned gases

Burned gases

2

Combustion wave fixed in the laboratory frame.

The fact that subsonic and supersonic waves can be obtained under almost the same conditions suggests that more can be learned by regarding the phenomena as overall fluid-mechanical in nature. Consider that the wave propagating in the tube is opposed by the unburned gases flowing at a velocity exactly equal to the wave propagation velocity. The wave then becomes fixed with respect to the containing tube (Fig. 4.1). This description of wave phenomena is readily treated analytically by the following integrated conservation equations, where the subscript 1 specifies the unburned gas conditions and subscript 2 the burned gas conditions:

c pT1 

ρ1u1  ρ2 u2

continuity

(4.1)

P1  ρ1u12  P2  ρ2 u22

momentum

(4.2)

energy

(4.3)

P1  ρ1 RT1

state

(4.4)

P2  ρ2 RT2

state

(4.5)

1 2 1 u1  q  c pT2  u22 2 2

Equation (4.4), which connects the known variables, unburned gas pressure, temperature, and density, is not an independent equation. In the coordinate system chosen, u1 is the velocity fed into the wave and u2 is the velocity coming out of the wave. In the laboratory coordinate system, the velocity ahead of the wave is zero, the wave velocity is u1, and (u1  u2) is the velocity of the burned gases with respect to the tube. The unknowns in the system are ul, u2, P2, T2, and ρ2. The chemical energy release is q, and the stagnation adiabatic combustion temperature is T2 for u2  0. The symbols follow the normal convention. Notice that there are five unknowns and only four independent equations. Nevertheless, one can proceed by analyzing the equations at hand. Simple algebraic manipulations (detailed in Chapter 5) result in two new equations: ⎛1 P⎞ 1 γ ⎛⎜ P2 1⎞ ⎜⎜ − 1 ⎟⎟⎟  ( P2  P1 ) ⎜⎜⎜  ⎟⎟⎟  q ⎜⎝ ρ1 ρ2 ⎟⎠ γ  1 ⎜⎝ ρ2 ρ1 ⎟⎠ 2

(4.6)

149

A J P2

Detonation

Flame Phenomena in Premixed Combustible Gases

I B III E

C

D II Deflagration

P1, 1/ρ1

K 1/ρ2

FIGURE 4.2 Reacting system (q 0) Hugoniot plot divided into five regimes A–E.

and

γM12 

⎛P ⎞ ⎜⎜ 2  1⎟⎟ ⎟⎟ ⎜⎜⎝ P ⎠ 1 ⎡ ⎤ ⎢1  (1/ρ2 ) ⎥ ⎢ (1/ρ1 ) ⎥⎦ ⎣

(4.7)

where γ is the ratio of specific heats and M is the wave velocity divided by (γRT1)1/2, the Mach number of the wave. For simplicity, the specific heats are assumed constant (i.e., cpl  cp2); however, γ is a much milder function of composition and temperature and the assumption that γ does not change between burned and unburned gases is an improvement. Equation (4.6) is referred to as the Hugoniot relationship, which states that for given initial conditions (Pl, 1/ρ1, q) a whole family of solutions (P2, 1/ρ2) is possible. The family of solutions lies on a curve on a plot of P2 versus 1/ρ2, as shown in Fig. 4.2. Plotted on the graph in Fig. 4.2 is the initial point (Pl, 1/ρl) and the two tangents through this point of the curve representing the family of solutions. One obtains a different curve for each fractional value of q. Indeed, a curve is obtained for q  0, that is, no energy release. This curve traverses the point, representing the initial condition and, since it gives the solution for simple shock waves, is referred to as the shock Hugoniot. Horizontal and vertical lines are drawn through the initial condition point, as well. These lines, of course, represent the conditions of constant pressure and constant specific volume (1/ρ), respectively. They further break the curve into three sections. Sections I and II are further divided into sections by the tangent points (J and K) and the other letters defining particular points. Examination of Eq. (4.7) reveals the character of Ml for regions I and II. In region I, P2 is much greater than P1, so that the difference is a number much larger than 1. Furthermore, in this region (1/ρ2) is a little less than (1/ρ1), and thus the ratio is a number close to, but a little less than 1. Therefore, the

150

Combustion

denominator is very small, much less than 1. Consequently, the right-hand side of Eq. (4.7) is positive and very much larger than 1, certainly greater than 1.4. If one assumes conservatively that γ  1.4, then M12 and M1 are both greater than 1. Thus, region I defines supersonic waves and is called the detonation region. Consequently, a detonation can be defined as a supersonic wave supported by energy release (combustion). One can proceed similarly in region II. Since P2 is a little less than P1, the numerator of Eq. (4.7) is a small negative fraction. (1/ρ2) is much greater than (1/ρ1), and so the denominator is a negative number whose absolute value is greater than 1. The right-hand side of Eq. (4.7) for region II is less than 1; consequently, Ml is less than 1. Thus region II defines subsonic waves and is called the deflagration region. Thus deflagration waves in this context are defined as subsonic waves supported by combustion. In region III, P2 P1 and 1/ρ2 1/ρ1, the numerator of Eq. (4.7) is positive, and the denominator is negative. Thus Ml is imaginary in region III and therefore does not represent a physically real solution. It will be shown in Chapter 5 that for points on the Hugoniot curve higher than J, the velocity of sound in the burned gases is greater than the velocity of the detonation wave relative to the burned gases. Consequently, in any real physical situation in a tube, wall effects cause a rarefaction. This rarefaction wave will catch up to the detonation front, reduce the pressure, and cause the final values of P2 and 1/ρ2 to drop to point J, the so-called Chapman–Jouguet point. Points between J and B are eliminated by considerations of the structure of the detonation wave or by entropy. Thus, the only steady-state solution in region I is given by point J. This unique solution has been found strictly by fluid-dynamic and thermodynamic considerations. Furthermore, the velocity of the burned gases at J and K can be shown to equal the velocity of sound there; thus M2  1 is a condition at both J and K. An expression similar to Eq. (4.7) for M2 reveals that M2 is greater than 1 as values past K are assumed. Such a condition cannot be real, for it would mean that the velocity of the burned gases would increase by heat addition, which is impossible. It is well known that heat addition cannot increase the flow of gases in a constant area duct past the sonic velocity. Thus region KD is ruled out. Unfortunately, there are no means by which to reduce the range of solutions that is given by region CK. In order to find a unique deflagration velocity for a given set of initial conditions, another equation must be obtained. This equation, which comes about from the examination of the structure of the deflagration wave, deals with the rate of chemical reaction or, more specifically, the rate of energy release. The Hugoniot curve shows that in the deflagration region the pressure change is very small. Indeed, approaches seeking the unique deflagration velocity assume the pressure to be constant and eliminate the momentum equation. The gases that flow in a Bunsen tube are laminar. Since the wave created in the horizontal tube experiment is so very similar to the Bunsen flame, it too is

Flame Phenomena in Premixed Combustible Gases

151

laminar. The deflagration velocity under these conditions is called the laminar flame velocity. The subject of laminar flame propagation is treated in the remainder of this section. For those who have not studied fluid mechanics, the definition of a deflagration as a subsonic wave supported by combustion may sound over sophisticated; nevertheless, it is the only precise definition. Others describe flames in a more relative context. A flame can be considered a rapid, self-sustaining chemical reaction occurring in a discrete reaction zone. Reactants may be introduced into this reaction zone, or the reaction zone may move into the reactants, depending on whether the unburned gas velocity is greater than or less than the flame (deflagration) velocity.

B. LAMINAR FLAME STRUCTURE Much can be learned by analyzing the structure of a flame in more detail. Consider, for example, a flame anchored on top of a single Bunsen burner as shown in Fig. 4.3. Recall that the fuel gas entering the burner induces air into the tube from its surroundings. As the fuel and air flow up the tube, they mix and, before the top of the tube is reached, the mixture is completely homogeneous. The flow velocity in the tube is considered to be laminar and the velocity across the tube is parabolic in nature. Thus the flow velocity near the tube wall is very low. This low flow velocity is a major factor, together with heat losses to the burner rim, in stabilizing the flame at the top. The dark zone designated in Fig. 4.3 is simply the unburned premixed gases before they enter the area of the luminous zone where reaction and heat release take place. The luminous zone is less than 1 mm thick. More specifically, the luminous zone is that portion of the reacting zone in which the temperature is the highest; indeed, much of the reaction and heat release take place in this zone. The color of the luminous zone changes with fuel–air ratio. For hydrocarbon–air mixtures that are fuel-lean, a deep violet radiation due to excited CH radicals appears. When the mixture is fuel-rich, the green radiation found is due to excited C2 molecules. The high-temperature burned gases usually show a reddish glow, which arises from CO2 and water vapor radiation. When the mixture is adjusted to be very rich, an intense yellow radiation can appear. This radiation is continuous and attributable to the presence of the solid carbon particles. Although Planck’s black-body curve peaks in the infrared for the temperatures that normally occur in these flames, the response of the human eye favors the yellow part of the electromagnetic spectrum. However, non-carbon-containing hydrogen–air flames are nearly invisible. Building on the foundation of the hydrocarbon oxidation mechanisms developed earlier, it is possible to characterize the flame as consisting of three zones [1]: a preheat zone, a reaction zone, and a recombination zone. The general structure of the reaction zone is made up of early pyrolysis reactions and a zone in which the intermediates, CO and H2, are consumed. For a very stable

152

Combustion

Burned gases Luminous zone Dark zone

Flow stream line

Premixed gas flow FIGURE 4.3

General description of laminar Bunsen burner flame.

molecule like methane, little or no pyrolysis can occur during the short residence time within the flame. But with the majority of the other saturated hydrocarbons, considerable degradation occurs, and the fuel fragments that leave this part of the reaction zone are mainly olefins, hydrogen, and the lower hydrocarbons. Since the flame temperature of the saturated hydrocarbons would also be very nearly the same (for reasons discussed in Chapter 1), it is not surprising that their burning velocities, which are very dependent on reaction rate, would all be of the same order (45 cm/s for a stoichiometric mixture in air). The actual characteristics of the reaction zone and the composition changes throughout the flame are determined by the convective flow of unburned gases toward the flame zone and the diffusion of radicals from the high-temperature reaction zone against the convective flow into the preheat region. This diffusion is dominated by H atoms; consequently, significant amounts of HO2 form in the lower-temperature preheat region. Because of the lower temperatures in the preheat zone, reaction (3.21) proceeds readily to form the HO2 radicals. At these temperatures the chain branching step (H  O2 → OH  O) does not occur. The HO2 subsequently forms hydrogen peroxide. Since the peroxide does not dissociate at the temperatures in the preheat zone, it is convected into the reaction zone where it forms OH radicals at the higher temperatures that prevail there [2].

Flame Phenomena in Premixed Combustible Gases

153

Owing to this large concentration of OH relative to O and H in the early part of the reaction zone, OH attack on the fuel is the primary reason for the fuel decay. Since the OH rate constant for abstraction from the fuel is of the same order as those for H and O, its abstraction reaction must dominate. The latter part of the reaction zone forms the region where the intermediate fuel molecules are consumed and where the CO is converted to CO2. As discussed in Chapter 3, the CO conversion results in the major heat release in the system and is the reason the rate of heat release curve peaks near the maximum temperature. This curve falls off quickly because of the rapid disappearance of CO and the remaining fuel intermediates. The temperature follows a smoother, exponential-like rise because of the diffusion of heat back to the cooler gases. The recombination zone falls into the burned gas or post-flame zone. Although recombination reactions are very exothermic, the radicals recombining have such low concentrations that the temperature profile does not reflect this phase of the overall flame system. Specific descriptions of hydrocarbon– air flames are shown later in this chapter.

C. THE LAMINAR FLAME SPEED The flame velocity—also called the burning velocity, normal combustion velocity, or laminar flame speed—is more precisely defined as the velocity at which unburned gases move through the combustion wave in the direction normal to the wave surface. The initial theoretical analyses for the determination of the laminar flame speed fell into three categories: thermal theories, diffusion theories, and comprehensive theories. The historical development followed approximately the same order. The thermal theories date back to Mallard and Le Chatelier [3], who proposed that it is propagation of heat back through layers of gas that is the controlling mechanism in flame propagation. As one would expect, a form of the energy equation is the basis for the development of the thermal theory. Mallard and Le Chatelier postulated (as shown in Fig. 4.4) that a flame consists of two zones separated at the point where the next layer ignites. Unfortunately, this thermal theory requires the concept of an ignition temperature. But adequate means do not exist for the determination of ignition temperatures; moreover, an actual ignition temperature does not exist in a flame. Later, there were improvements in the thermal theories. Probably the most significant of these is the theory proposed by Zeldovich and Frank-Kamenetskii. Because their derivation was presented in detail by Semenov [4], it is commonly called the Semenov theory. These authors included the diffusion of molecules as well as heat, but did not include the diffusion of free radicals or atoms. As a result, their approach emphasized a thermal mechanism and was widely used in correlations of experimental flame velocities. As in the

154

Combustion

T

Tf

Ti

Zone II Region of conduction δ T0

Zone I

Region of burning X

FIGURE 4.4

Mallard–Le Chatelier description of the temperature in a laminar flame wave.

Mallard–Le Chatelier theory, Semenov assumed an ignition temperature, but by approximations eliminated it from the final equation to make the final result more useful. This approach is similar to what is now termed activation energy asymptotics. The theory was advanced further when it was postulated that the reaction mechanism can be controlled not only by heat, but also by the diffusion of certain active species such as radicals. As described in the preceding section, low-atomic- and molecular-weight particles can readily diffuse back and initiate further reactions. The theory of particle diffusion was first advanced in 1934 by Lewis and von Elbe [5] in dealing with the ozone reaction. Tanford and Pease [6] carried this concept further by postulating that it is the diffusion of radicals that is all important, not the temperature gradient as required by the thermal theories. They proposed a diffusion theory that was quite different in physical concept from the thermal theory. However, one should recall that the equations that govern mass diffusion are the same as those that govern thermal diffusion. These theories fostered a great deal of experimental research to determine the effect of temperature and pressure on the flame velocity and thus to verify which of the theories were correct. In the thermal theory, the higher the ambient temperature, the higher is the final temperature and therefore the faster is the reaction rate and flame velocity. Similarly, in the diffusion theory, the higher the temperature, the greater is the dissociation, the greater is the concentration of radicals to diffuse back, and therefore the faster is the velocity. Consequently, data obtained from temperature and pressure effects did not give conclusive results. Some evidence appeared to support the diffusion concept, since it seemed to best explain the effect of H2O on the experimental flame velocities of CO¶O2. As described in the previous chapter, it is known that at high temperatures water provides the source of hydroxyl radicals to facilitate rapid reaction of CO and O2.

Flame Phenomena in Premixed Combustible Gases

155

Hirschfelder et al. [7] reasoned that no dissociation occurs in the cyanogen– oxygen flame. In this reaction the products are solely CO and N2, no intermediate species form, and the C¨O and N˜N bonds are difficult to break. It is apparent that the concentration of radicals is not important for flame propagation in this system, so one must conclude that thermal effects predominate. Hirschfelder et al. [7] essentially concluded that one should follow the thermal theory concept while including the diffusion of all particles, both into and out of the flame zone. In developing the equations governing the thermal and diffusional processes, Hirschfelder obtained a set of complicated nonlinear equations that could be solved only by numerical methods. In order to solve the set of equations, Hirschfelder had to postulate some heat sink for a boundary condition on the cold side. The need for this sink was dictated by the use of the Arrhenius expressions for the reaction rate. The complexity is that the Arrhenius expression requires a finite reaction rate even at x  , where the temperature is that of the unburned gas. In order to simplify the Hirschfelder solution, Friedman and Burke [8] modified the Arrhenius reaction rate equation so the rate was zero at T  T0, but their simplification also required numerical calculations. Then it became apparent that certain physical principles could be used to simplify the complete equations so they could be solved relatively easily. Such a simplification was first carried out by von Karman and Penner [9]. Their approach was considered one of the more significant advances in laminar flame propagation, but it could not have been developed and verified if it were not for the extensive work of Hirschfelder and his collaborators. The major simplification that von Karman and Penner introduced is the fact that the eigenvalue solution of the equations is the same for all ignition temperatures, whether it be near Tf or not. More recently, asymptotic analyses have been developed that provide formulas with greater accuracy and further clarification of the wave structure. These developments are described in detail in three books [10–12]. It is easily recognized that any exact solution of laminar flame propagation must make use of the basic equations of fluid dynamics modified to account for the liberation and conduction of heat and for changes of chemical species within the reaction zones. By use of certain physical assumptions and mathematical techniques, the equations have been simplified. Such assumptions have led to many formulations (see Refs. [10–12]), but the theories that will be considered here are an extended development of the simple Mallard–Le Chatelier approach and the Semenov approach. The Mallard–Le Chatelier development is given because of its historical significance and because this very simple thermal analysis readily permits the establishment of the important parameters in laminar flame propagation that are more difficult to interpret in the complex analyses. The Zeldovich–Frank-Kamenetskii–Semenov theory is reviewed because certain approximations related to the ignition temperature that are employed are useful in other problems in the combustion field and permit an introductory understanding to activation energy asymptotics.

156

Combustion

1. The Theory of Mallard and Le Chatelier Conceptually, Mallard and Le Chatelier stated that the heat conducted from zone II in Fig. 4.4 is equal to that necessary to raise the unburned gases to the ignition temperature (the boundary between zones I and II). If it is assumed that the slope of the temperature curve is linear, the slope can be approximated by the expression [(Tf  Ti)/δ], where Tf is the final or flame temperature, Ti is the ignition temperature, and δ is the thickness of the reaction zone. The enthalpy balance then becomes  p (Ti  T0 )  λ mc

(Tf − Ti ) A δ

(4.8)

where λ is the thermal conductivity, m is the mass rate of the unburned gas mixture into the combustion wave, T0 is the temperature of the unburned gases, and A is the cross-sectional area taken as unity. Since the problem as described is fundamentally one-dimensional, m  ρ Au  ρ SL A

(4.9)

where ρ is the density, u is the velocity of the unburned gases, and SL is the symbol for the laminar flame velocity. Because the unburned gases enter normal to the wave, by definition SL  u

(4.10)

ρ SL c p (Ti  T0 )  λ(Tf  Ti )/δ

(4.11)

⎛ λ(T  T ) 1 ⎞⎟ ⎜ f i ⎟⎟ SL  ⎜⎜ ⎜⎝ ρ c p (Ti  T0 ) δ ⎟⎟⎠

(4.12)

Equation (4.8) then becomes

or

Equation (4.12) is the expression for the flame speed obtained by Mallard and Le Chatelier. Unfortunately, in this expression δ is not known; therefore, a better representation is required. Since δ is the reaction zone thickness, it is possible to relate δ to SL. The total rate of mass per unit area entering the reaction zone must be the mass rate of consumption in that zone for the steady flow problem being considered. Thus  m /A  ρu  ρ SL  ωδ

(4.13)

Flame Phenomena in Premixed Combustible Gases

157

where ω specifies the reaction rate in terms of concentration (in grams per cubic centimeter) per unit time. Equation (4.12) for the flame velocity then becomes ⎡ λ (T  T ) ω ⎤1/ 2 ⎛ ω ⎞1/ 2 f i ⎥ ∼ ⎜⎜ α ⎟⎟ SL  ⎢⎢ ⎥ ⎜⎝ ρ ⎟⎟⎠ ρ c ( T  T ) ρ ⎢⎣ p i ⎥⎦ 0

(4.14)

where it is important to understand that ρ is the unburned gas density and α is the thermal diffusivity. More fundamentally the mass of reacting fuel mixture consumed by the laminar flame is represented by ⎛ λ ⎞⎟1/ 2 ⎜ ρ SL ∼ ⎜⎜ ω ⎟⎟⎟ ⎜⎝ c p ⎟⎠

(4.15)

Combining Eqs. (4.13) and (4.15), one finds that the reaction thickness in the complete flame wave is δ ∼ α /SL

(4.16)

This adaptation of the simple Mallard–Le Chatelier approach is most significant in that the result ⎛ ω ⎞1/ 2 SL ∼ ⎜⎜⎜α ⎟⎟⎟ ⎝ ρ ⎟⎠ is very useful in estimating the laminar flame phenomena as various physical and chemical parameters are changed. Linan and Williams [13] review the description of the flame wave offered by Mikhelson [14], who equated the heat release in the reaction zone to the conduction of energy from the hot products to the cool reactants. Since the overall conservation of energy shows that the energy per unit mass (h) added to the mixture by conduction is h  c p (Tf  T0 )

(4.17)

 L  λ(Tf  T0 )/δL hωδ

(4.18)

then

In this description δL represents not only the reaction zone thickness δ in the Mallard–Le Chatelier consideration, but also the total of zones I and II in Fig. 4.4. Substituting Eq. (4.17) into Eq. (4.18) gives  L  λ(Tf  T0 )/δL c p (Tf  T0 )ωδ

158

Combustion

or ⎛ λ 1 ⎞⎟1/ 2 ⎜ ⎟⎟ δL  ⎜⎜ ⎜⎝ c p ω ⎟⎟⎠ The conditions of Eq. (4.13) must hold, so that in this case  L ρ SL  ωδ and Eq. (4.18) becomes 1/ 2 ⎛ λ ω ⎞⎟1/ 2 ⎛⎜ ω ⎞⎟ SL  ⎜⎜ ⎟⎟  ⎜⎜ α ⎟⎟ ⎜⎜ ρ c p ρ ⎟⎠ ⎝ ρ ⎟⎠ ⎝

(4.19)

Whereas the proportionality of Eq. (4.14) is the same as the equality in Eq. (4.19), the difference in the two equations is the temperature ratio ⎛ T  T ⎞⎟1/ 2 i ⎟ ⎜⎜ f ⎜⎜ T  T ⎟⎟ ⎝ i 0⎠ In the next section, the flame speed development of Zeldovich, FrankKamenetskii, and Semenov will be discussed. They essentially evaluate this term to eliminate the unknown ignition temperature Ti by following what is now the standard procedure of narrow reaction zone asymptotics, which assumes that the reaction rate decreases very rapidly with a decrease in temperature. Thus, in the course of the integration of the rate term ω in the reaction zone, they extend the limits over the entire flame temperature range T0 to Tf. This approach is, of course, especially valid for large activation energy chemical processes, which are usually the norm in flame studies. Anticipating this development, one sees that the temperature term essentially becomes RTf2 E (Tf  T0 ) This term specifies the ratio δL/δ and has been determined explicitly by Linan and Williams [13] by the procedure they call activation energy asymptotics. Essentially, this is the technique used by Zeldovich, Frank-Kamenetskii, and Semenov [see Eq. (4.59)]. The analytical development of the asymptotic approach is not given here. For a discussion of the use of asymptotics, one should refer to the excellent books by Williams [12], Linan and Williams [13], and Zeldovich et al. [10]. Linan and Williams have called the term

Flame Phenomena in Premixed Combustible Gases

159

RTf2 /E (Tf  T0 ) the Zeldovich number and give this number the symbol β in their book. Thus β  (δL /δ ) It follows, then, that Eq. (4.14) may be rewritten as ⎛ α ω ⎞⎟1/ 2 ⎟ SL  ⎜⎜ ⎜⎝ β ρ ⎟⎟⎠

(4.20)

and, from the form of Eq. (4.13), that δL  βδ 

αβ SL

(4.21)

The general range of hydrocarbon–air premixed flame speeds falls around 40 cm/s. Using a value of thermal diffusivity evaluated at a mean temperature of 1300 K, one can estimate δL to be close to 0.1 cm. Thus, hydrocarbon–air flames have a characteristic length of the order of 1 mm. The characteristic time is (α/SL2), and for these flames this value is estimated to be of the order of a few milliseconds. If one assumes that the overall activation energy of the hydrocarbon–air process is of the order 160 kJ/mol and that the flame temperature is 2100 K, then β is about 10, and probably somewhat less in actuality. Thus, it is estimated from this simple physical approach that the reaction zone thickness, δ, would be a small fraction of a millimeter. The simple physical approaches proposed by Mallard and Le Chatelier [3] and Mikhelson [14] offer significant insight into the laminar flame speed and factors affecting it. Modern computational approaches now permit not only the calculation of the flame speed, but also a determination of the temperature profile and composition changes throughout the wave. These computational approaches are only as good as the thermochemical and kinetic rate values that form their database. Since these approaches include simultaneous chemical rate processes and species diffusion, they are referred to as comprehensive theories, which is the topic of Section C3. Equation (4.20) permits one to establish various trends of the flame speed as various physical parameters change. Consider, for example, how the flame speed should change with a variation of pressure. If the rate term ω follows second-order kinetics, as one might expect from a hydrocarbon–air system, then the concentration terms in ω would be proportional to P2. However, the density term in α(λ/ρcp) and the other density term in Eq. (4.20) also give a P2 dependence. Thus for a second-order reaction system the flame speed appears independent of pressure. A more general statement of the pressure

160

Combustion

dependence in the rate term is that ω  Pn, where n is the overall order of the reaction. Thus it is found that SL ∼ ( P n2 )1/ 2

(4.22)

For a first-order dependence such as that observed for a hydrazine decomposition flame, SL  P1/2. As will be shown in Section C5, although hydrocarbon– air oxidation kinetics are approximately second-order, many hydrocarbon–air flame speeds decrease as the pressure rises. This trend is due to the increasing role of the third-order reaction H  O2  M → HO2  M in effecting the chain branching and slowing the rate of energy release. Although it is now realized that SL in these hydrocarbon systems may decrease with pressure, it is important to recognize that the mass burning rate ρSL increases with pressure. Essentially, then, one should note that m 0 ≡ ρ SL ∼ P n/ 2

(4.23)

where m 0 is the mass flow rate per unit area of the unburned gases. Considering β a constant, the flame thickness δL decreases as the pressure rises since δL ∼

α λ λ ∼ ∼ SL cp ρ SL c p m 0

(4.24)

Since (λ/cp) does not vary with pressure and m 0 increases with pressure as specified by Eq. (4.23), then Eq. (4.24) verifies that the flame thickness must decrease with pressure. It follows from Eq. (4.24) as well that λ m 0δL ∼ c p

(4.25)

or that m 0δL is essentially equal to a constant, and that for most hydrocarbon– air flames in which nitrogen is the major species and the reaction product molar compositions do not vary greatly, m 0δL is the same. How these conclusions compare with the results of comprehensive theory calculations will be examined in Section C5. The temperature dependence in the flame speed expression is dominated by the exponential in the rate expression ω ; thus, it is possible to assume that 1/ 2 SL ∼ ⎡⎣ exp(E/RT ) ⎤⎦

(4.26)

The physical reasoning used indicates that most of the reaction and heat release must occur close to the highest temperature if high activation energy

161

Flame Phenomena in Premixed Combustible Gases

Arrhenius kinetics controls the process. Thus the temperature to be used in the above expression is Tf and one rewrites Eq. (4.26) as 1/ 2 SL ∼ ⎡⎣ exp(E/RTf ) ⎤⎦

(4.27)

Thus, the effect of varying the initial temperature is found in the degree to which it alters the flame temperature. Recall that, due to chemical energy release, a 100° rise in initial temperature results in a rise of flame temperature that is much smaller. These trends due to temperature have been verified experimentally.

2. The Theory of Zeldovich, Frank-Kamenetskii, and Semenov As implied in the previous section, the Russian investigators Zeldovich, Frank-Kamenetskii, and Semenov derived an expression for the laminar flame speed by an important extension of the very simplified Mallard–Le Chatelier approach. Their basic equation included diffusion of species as well as heat. Since their initial insight was that flame propagation was fundamentally a thermal mechanism, they were not concerned with the diffusion of radicals and its effect on the reaction rate. They were concerned with the energy transported by the diffusion of species. As in the Mallard–Le Chatelier approach, an ignition temperature arises in this development, but it is used only as a mathematical convenience for computation. Because the chemical reaction rate is an exponential function of temperature according to the Arrhenius equation, Semenov assumed that the ignition temperature, above which nearly all reaction occurs, is very near the flame temperature. With this assumption, the ignition temperature can be eliminated in the mathematical development. Since the energy equation is the one to be solved in this approach, the assumption is physically correct. As described in the previous section for hydrocarbon flames, most of the energy release is due to CO oxidation, which takes place very late in the flame where many hydroxyl radicals are available. For the initial development, although these restrictions are partially removed in further developments, two other important assumptions are made. The assumptions are that the cp and λ are constant and that (λ / c p )  D ρ where D is the mass diffusivity. This assumption is essentially that αD Simple kinetic theory of gases predicts αDν

162

Combustion

where ν is kinematic viscosity (momentum diffusivity). The ratios of these three diffusivities give some of the familiar dimensionless similarity parameters, Pr  ν /α,

Sc  ν /D,

Le  α /D

where Pr, Sc, and Le are the Prandtl, Schmidt, and Lewis numbers, respectively. The Prandtl number is the ratio of momentum to thermal diffusion, the Schmidt number is momentum to mass diffusion, and the Lewis number is thermal to mass diffusion. Elementary kinetic theory of gases then predicts as a first approximation Pr  Sc  Le  1 With this approximation, one finds (λ / c p )  D ρ ≠ f (P ) that is, neither (λ/cp) nor Dρ is a function of pressure. Consider the thermal wave given in Fig. 4.4. If a differential control volume is taken within this one-dimensional wave and the variations as given in the figure are in the x direction, then the thermal and mass balances are as shown in Fig. 4.5. In Fig. 4.5, a is the mass of reactant per cubic centimeter, ω is the rate of reaction, Q is the heat of reaction per unit mass, and ρ is the total density. Note that a/ρ is the mass fraction of reactant a. Since the problem is a steady one, there is no accumulation of species or heat with respect to time, and the balance of the energy terms and the species terms must each be equal to zero.

T

T

( dT ) dx

ω Q

(a /ρ) 

d(a/ρ)

x dx d(a/ρ) x m (a/ρ)  dx d(a/ρ) d (a/ρ)  x Dρ dx dx

(a/ρ)

[

m (a/ρ) Dρ

dT x ) dx dT d x ) λ (T  dx dx m Cp (T 

m Cp T λ

dT x dx

]

[

d(a/ρ) dx

]

Δx FIGURE 4.5 Balances across a differential element in a thermal wave describing a laminar flame.

163

Flame Phenomena in Premixed Combustible Gases

The amount of mass convected into the volume AΔx (where A is the area usually taken as unity) is ⎡⎛ a ⎞ d (a/ρ) ⎤ ⎛a⎞ d (a/ρ) AΔx m ⎢⎢⎜⎜ ⎟⎟⎟  Δx ⎥⎥ A  m ⎜⎜⎜ ⎟⎟⎟ A  m ⎟ ⎟ ⎜ dx dx ⎝ρ⎠ ⎢⎣⎝ ρ ⎠ ⎥⎦

(4.28)

For this one-dimensional configuration m  ρ0SL. The amount of mass diffusing into the volume is



d dx

⎡ ⎞⎤ ⎛ ⎛ ⎢ Dρ ⎜⎜ a  d (a/ρ) Δx ⎟⎟⎥ A  ⎜⎜Dρ d (a/ρ) ⎟ ⎢ ⎥ ⎜⎝ ρ ⎜ ⎟⎠⎥ dx ρ dx ⎝ ⎢⎣ ⎦

⎞⎟ d 2 (a/ρ ) ⎟⎟ A  (Dρ) AΔx dx 2 ⎠⎟ (4.29)

The amount of mass reacting (disappearing) in the volume is ω A Δx and it is to be noted that ω is a negative quantity. Thus the continuity equation for the reactant is d 2 (a/ρ) dx 2 (diffusion term)

(Dρ)



d (a /ρ) dx (convective term) m

ω  0



(generation term) (4.30)

The energy equation is determined similarly and is λ

d 2T dT  p  mc  ω Q  0 dx 2 dx

(4.31)

Because ω is negative and the overall term must be positive since there is heat release, the third term has a negative sign. The state equation is written as (ρ /ρ0 )  (T0 /T ) New variables are defined as T 

c p (T  T0 )

Q a  (a0 /ρ0 )  (a/ρ )

164

Combustion

where the subscript 0 designates initial conditions. Substituting the new variables in Eqs. (4.30) and (4.31), one obtains two new equations: d 2 a da  m  ω  0 2 dx dx

(4.32)

λ d 2T dT   m  ω  0 c p dx 2 dx

(4.33)

Dρ

The boundary conditions for these equations are x  ∞, x  ∞,

a  0, a  a0 /ρ0,

T  0 T  [c p (Tf  T0 )]/Q

(4.34)

where Tf is the final or flame temperature. For the condition Dρ  (λ/cp), Eqs. (4.32) and (4.33) are identical in form. If the equations and boundary conditions for a and T coincide; that is, if a  T over the entire interval, then c pT0  (a0Q/ρ0 )  c pTf  c pT  (aQ/ρ)

(4.35)

The meaning of Eq. (4.35) is that the sum of the thermal and chemical energies per unit mass of the mixture is constant in the combustion zone; that is, the relation between the temperature and the composition of the gas mixture is the same as that for the adiabatic behavior of the reaction at constant pressure. Thus, the variable defined in Eq. (4.35) can be used to develop a new equation in the same manner as Eq. (4.30), and the problem reduces to the solution of only one differential equation. Indeed, either Eq. (4.30) or (4.31) can be solved; however, Semenov chose to work with the energy equation. In the first approach it is assumed, as well, that the reaction proceeds by zero-order. Since the rate term ω is not a function of concentration, the continuity equation is not required so we can deal with the more convenient energy equation. Semenov, like Mallard and Le Chatelier, examined the thermal wave as if it were made up of two parts. The unburned gas part is a zone of no chemical reaction, and the reaction part is the zone in which the reaction and diffusion terms dominate and the convective term can be ignored. Thus, in the first zone (I), the energy equation reduces to  p dT mc d 2T  0 dx 2 λ dx

(4.36)

with the boundary conditions x  ∞,

T  T0 ;

x  0,

T  Ti

(4.37)

165

Flame Phenomena in Premixed Combustible Gases

It is apparent from the latter boundary condition that the coordinate system is so chosen that the Ti is at the origin. The reaction zone extends a small distance δ, so that in the reaction zone (II) the energy equation is written as d 2T ω Q  0 2 dx λ

(4.38)

with the boundary conditions x  0,

T  Ti ;

x  δ,

T  Tf

The added condition, which permits the determination of the solution (eigenvalue), is the requirement of the continuity of heat flow at the interface of the two zones: ⎛ dT ⎞⎟ ⎛ dT ⎞⎟ λ ⎜⎜  λ ⎜⎜ ⎟ ⎟ ⎜⎝ dx ⎟⎠ ⎜⎝ dx ⎟⎠ x0,I x0,II

(4.39)

The solution to the problem is obtained by initially considering Eq. (4.38). First, recall that 2 ⎛ dT ⎞⎟ d 2T d ⎛⎜ dT ⎞⎟ ⎟⎟  2 ⎜⎜ ⎟ ⎜⎜ ⎜⎝ dx ⎟⎠ dx 2 dx ⎝ dx ⎠

(4.40)

Now, Eq. (4.38) is multiplied by 2 (dT/dx) to obtain ⎛ dT ⎞⎟ d 2T ω Q ⎛⎜ dT ⎞⎟  2 2 ⎜⎜ ⎟ ⎟ ⎜ ⎜⎝ dx ⎟⎠ dx 2 λ ⎜⎝ dx ⎟⎠

(4.41)

2 d ⎛⎜ dT ⎞⎟ ω Q ⎛⎜ dT ⎞⎟  2 ⎟ ⎟ ⎜⎜ ⎜ ⎟ dx ⎝ dx ⎠ λ ⎜⎝ dx ⎟⎠

(4.42)

Integrating Eq. (4.42), one obtains ⎛ dT ⎞⎟2 Q  ⎜⎜  2 ⎟ ⎜⎝ dx ⎟⎠ λ x 0

Tf

∫ ω dT

(4.43)

Ti

since (dT/dx)2, evaluated at x  δ or T  Tf, is equal to zero. But from Eq. (4.36), one has  p /λ )T  const dT/dx  (mc

(4.44)

166

Combustion

Since at x  , T  T0 and (dT/dx)  0,  p /λ )T0 const  (mc

(4.45)

 p (T  T0 )]/λ dT/dx  [ mc

(4.46)

and

Evaluating the expression at x  0 where T  Ti, one obtains  p (Ti  T0 )/λ (dT/dx ) x0  mc

(4.47)

The continuity of heat flux permits this expression to be combined with Eq. (4.43) to obtain  p (Ti  T0 ) mc λ

⎛ ⎜ 2Q  ⎜⎜⎜ ⎜⎝ λ

Tf

∫ Ti

⎞⎟1/ 2 ω dT ⎟⎟⎟ ⎟⎟ ⎠

Since Arrhenius kinetics dominate, it is apparent that Ti is very close to Tf, so the last expression is rewritten as  p (Tf − T0 ) mc λ

⎛ ⎜ 2Q  ⎜⎜⎜ ⎜⎝ λ

Tf

∫ Ti

⎞⎟1/ 2 ω dT ⎟⎟⎟ ⎟⎟ ⎠

(4.48)

For m  SL ρ0 and (a0/ρ0)Q taken equal to cp(Tf  T0) [from Eq. (4.35)], one obtains ⎡ ⎛ ⎤ 1/ 2 ⎞ I ⎜⎜ λ ⎟⎟ ⎢ ⎥ SL  ⎢ 2 ⎜ ⎟ ⎥ ⎢⎣ ⎜⎝ ρ c p ⎟⎟⎠ (Tf  T0 ) ⎥⎦

(4.49)

where I

1 a0

Tf

∫ ω dT

(4.50)

Ti

and ω is a function of T and not of concentration for a zero-order reaction. Thus it may be expressed as ω  Z  eE/RT

(4.51)

167

Flame Phenomena in Premixed Combustible Gases

where Z is the pre-exponential term in the Arrhenius expression. For sufficiently large energy of activation such as that for hydrocarbon– oxygen mixtures where E is of the order of 160 kJ/mol, (E/RT) 1. Thus most of the energy release will be near the flame temperature, Ti will be very near the flame temperature, and zone II will be a very narrow region. Consequently, it is possible to define a new variable σ such that σ  (Tf  T )

(4.52)

σi  (Tf  Ti )

(4.53)

The values of σ will vary from

to zero. Since σ Tf then (E/RT )  ⎡⎣ E/R(Tf  σ ) ⎤⎦  ⎡⎣ E/RTf (1  σ /Tf ) ⎤⎦  (E/RTf ) ⎡⎣1  (σ /Tf ) ⎤⎦  (E/RTf )  (E σ /RTf2 ) Thus the integral I becomes I

Z  eE/RTf a0

Tf

∫

eE σ /RTf dT  2

Ti

Z  eE/RTf a0

0

∫ eEσ /RT

2 f

dσ

(4.54)

σi

Defining still another variable β as β  E σ /RTf2

(4.55)

the integral becomes I

Z  eE/RTf a0

⎞⎟ ⎛ βi 2 ⎜⎜ β ⎟⎟ RTf e d β ⎜⎜ ∫ ⎟⎟ E ⎜⎝ 0 ⎟⎠

(4.56)

With sufficient accuracy one may write βi

j  ∫ eβ d β  (1  eβi ) ≅ 1

(4.57)

0

since (E/RTf) 1 and (σi/Tf)  0.25. Thus, ⎛ Z  ⎞ ⎛ RT 2 ⎞ I  ⎜⎜⎜ ⎟⎟⎟ ⎜⎜⎜ f ⎟⎟⎟ eE/RTf ⎜⎝ a0 ⎟⎠ ⎜⎝ E ⎟⎠

(4.58)

168

Combustion

and ⎡ 2 SL  ⎢⎢ ⎢⎣ a0

1/ 2 ⎛ λ ⎞⎟ ⎞⎟⎤⎥ ⎛ RTf2 ⎜⎜ ⎟⎟ ( Z  eE/RTf ) ⎜⎜ ⎟ ⎜⎜ ⎜⎜ E (T   ) ⎟⎟⎥ ⎟ ⎝ f 0 ⎠⎥ ⎝ ρ0 c p ⎟⎠ ⎦

(4.59)

In the preceding development, it was assumed that the number of moles did not vary during reaction. This restriction can be removed to allow the number to change in the ratio (nr /np), which is the number of moles of reactant to product. Furthermore, the Lewis number equal to one restriction can be removed to allow (λ / c p )D ρ  A / B where A and B are constants. With these restrictions removed, the result for a first-order reaction becomes 2 ⎡ 2λ (c ) Z  ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎤ 1/ 2 ⎢ f p f ⎜ T0 ⎟ ⎜ nr ⎟ ⎜ A ⎟ ⎜ RTf2 ⎞⎟⎟ eE/RTf ⎥ SL  ⎢ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎥ ⎟ ⎢ ρ0 c p2 ⎝ Tf ⎟⎠ ⎜⎝ np ⎟⎠ ⎝ B ⎟⎠ ⎝ E ⎠ (Tf  T0 )2 ⎥ ⎣ ⎦

(4.60a)

and for a second-order reaction ⎡ 2λc 2 Z a 0 ⎢ pf SL  ⎢ ⎢ ρ0 (c p )3 ⎣

1/ 2 3 ⎛ T0 ⎞⎟2 ⎛ nr ⎞⎟ ⎛ A ⎞⎟2 ⎛⎜ RTf2 ⎞⎟ eE/RTf ⎤⎥ ⎜⎜ ⎟ ⎜⎜ ⎟ ⎜⎜ ⎟ ⎜ ⎟ ⎜⎝ T ⎟⎟⎠ ⎜⎜ n ⎟⎟ ⎜⎝ B ⎟⎟⎠ ⎜⎝ E ⎟⎠ (T  T )3 ⎥⎥ 0 f f ⎝ p⎠ ⎦

(4.60b)

where cpf is the specific heat evaluated at Tf and c p is the average specific heat between T0 and Tf. Notice that a0 and ρ0 are both proportional to pressure and SL is independent of pressure. Furthermore, this complex development shows that ⎛ λc 2 ⎞⎟1/ 2 pf ⎜⎜  E / RT ⎟⎟ , f SL ∼ ⎜ a Z e ⎟⎟ ⎜⎝ ρ0 (c p )3 0 ⎠

⎛ λ ⎞⎟1/ 2 ⎜⎜ SL ∼ ⎜ RR ⎟⎟⎟ ∼ (αRR)1/ 2 ⎜⎝ ρ0 c p ⎟⎠

(4.61)

as was obtained from the simple Mallard–Le Chatelier approach.

3. Comprehensive Theory and Laminar Flame Structure Analysis To determine the laminar flame speed and flame structure, it is now possible to solve by computational techniques the steady-state comprehensive mass, species, and energy conservation equations with a complete reaction mechanism for the fuel–oxidizer system which specifies the heat release. The numerical

169

Flame Phenomena in Premixed Combustible Gases

code for this simulation of a freely propagating, one-dimensional, adiabatic premixed flame is based on the scheme of Kee et al. [15]. The code uses a hybrid time–integration/Newton-iteration technique to solve the equations. The mass burning rate of the flame is calculated as an eigenvalue of the problem and, since the unburned gas mixture density is known, the flame speed SL is determined ( m  ρ0SL). In addition, the code permits one to examine the complete flame structure and the sensitivities of all reaction rates on the temperature and species profiles as well as on the mass burning rate. Generally, two preprocessors are used in conjunction with the freely propagating flame code. The first, CHEMKIN, is used to evaluate the thermodynamic properties of the reacting mixture and to process an established chemical reaction mechanism of the particular fuel–oxidizer system [16]. The second is a molecular property package that provides the transport properties of the mixture [17]. See Appendix I. In order to evaluate the flame structure of characteristic fuels, this procedure was applied to propane, methane, and hydrogen–air flames at the stoichiometric equivalence ratio and unburned gas conditions of 298 K and 1 atm. The fuels were chosen because of their different kinetic characteristics. Propane is characteristic of most of the higher-order hydrocarbons. As discussed in the previous chapter, methane is unique with respect to hydrocarbons, and hydrogen is a non-hydrocarbon that exhibits the largest mass diffusivity trait. Table 4.1 reports the calculated values of the mass burning rate and laminar flame speed, and Figs. 4.6–4.12 report the species, temperature, and heat release rate distributions. These figures and Table 4.1 reveal much about the flame structure and confirm much of what was described in this and preceding chapters. The δL reported in Table 4.1 was estimated by considering the spatial distance of the first perceptible rise of the temperature or reactant decay in the figures and the extrapolation of the q curve decay branch to the axis. This procedure eliminates the gradual curvature of the decay branch near the point where all fuel elements are consumed and which is due to radical recombination. Since

TABLE 4.1 Flame Properties at φ  1a m 0 δL (g/cm s)

SL (cm/s)

m 0  ρSL (g/cm2 s)

219.7

0.187

0.050 (Fig. 4.11)

0.0093

0.73

CH4

36.2

0.041

0.085 (Fig. 4.9)

0.0035

1.59

C3H8

46.3

0.055

0.057 (Fig. 4.6)

0.0031

1.41

Fuel– Air H2

T0  298 K, P  1 atm.

a

δL [cm (est.)]

( m 0δL)/(λ/cp)0

170

Combustion

2500

0.25 T O2

2000

H2O

0.15

0.10

0.050

0.0 0.0

q/5

CO2 H2  5

C3H8

CO

0.5

1.0

1.5

2.0

2.5

1500

1000

Temperature (K) or heat release rate (J/s cm3)

Mole fraction

0.20

500

0 3.0

Flame coordinate (mm) FIGURE 4.6 Composition, temperature, and heat release rate profiles for a stoichiometric C3H8–air laminar flame at 1 atm and T0  298 K. 0.012

2500 C3H8/4

T

0.010

2000

Mole fraction

1500 0.0060 1000

q/5 0.0040

C3H6 C2H6

0.0020

Temperature (K) or heat release rate (J/s cm3)

0.0080

500

CH4 C2H4

0.0 0.0 FIGURE 4.7

0.5

1.0

1.5

2.0

2.5

0 3.0

Flame coordinate (mm) Reaction intermediates for Fig. 4.6.

for hydrocarbons one would expect (λ/cp) to be approximately the same, the values of m 0 δL for CH4 and C3H8 in Table 4.1 should be quite close, as indeed they are. Since the thermal conductivity of H2 is much larger than that of gaseous hydrocarbons, it is not surprising that its value of m 0 δL is larger than

171

Flame Phenomena in Premixed Combustible Gases

0.012

2500 T

0.010

2000

Mole fraction

OH 1500

0.0060 H 1000

Temperature (K)

0.0080

0.0040 O

500

0.0020 HO2  10 0.0 0.0

CH2O

0.5

1.0

1.5

2.0

2.5

0 3.0

Flame coordinate (mm) FIGURE 4.8 Radical distribution profiles for Fig. 4.6. 2500

0.25 T 0.2

2000

Mole fraction

H2O 0.15

0.1

1500

CH4

1000

q/5

CO2

0.05

Temperature (K) or heat release rate (J/s cm3)

O2

500 CO

0 0.0

0.5

1.0

1.5

2.0

2.5

0 3.0

Flame coordinate (mm) FIGURE 4.9 Composition, temperature, and heat release rate profiles for a stoichiometric CH4–air laminar flame at 1 atm and T0  298 K.

those for CH4 and C3H8. What the approximation m 0 δL ∼ (λ /c p ) truly states is that m 0 δL /(λ /c p ) is of order 1. This order simply arises from the fact that if the thermal equation in the flame speed development is nondimensionalized with δL and SL as the critical dimension and velocity, then m 0 δL / (λ /c p ) is the

172

Combustion

2500

0.01 T 0.008

2000

Mole fraction

0.006

1500 H

0.004

1000

Temperature (K)

OH

O 0.002 HO2  10 0 0.0 0.5 FIGURE 4.10

500

CH3 CH2O

1.0 1.5 2.0 Flame coordinate (mm) Radical distribution profiles for Fig. 4.9.

2.5

0.4

0 3.0

2500 T

Mole fraction

2000

H2 H2O

1500 0.2 O2

1000

q/10

0.1

Temperature (K) or heat release rate (J/s cm3)

0.3

500

0

0.0

0.5

1.0

1.5

2.0

2.5

0 3.0

Flame coordinate (mm) FIGURE 4.11 Composition, temperature, and heat release rate profiles for a stoichiometric H2–air laminar flame at 1 atm and T0  298 K.

173

Flame Phenomena in Premixed Combustible Gases

0.04

2500 H

T 2000

1500 0.02 HO2  50

1000

OH

Temperature (K)

Mole fraction

0.03

0.01 500

O H2O2  50 0 0.0

0.5

1.0

1.5

2.0

2.5

0 3.0

Flame coordinate (mm) FIGURE 4.12 Radical distribution profiles for Fig. 4.11.

Peclet number (Pe) before the convection term in this equation. This point can be readily seen from ρδ S m 0 δL S δ  0 L L  L L  Pe (λ / c p ) (λ / c p ) α0 Since m 0  ρ0SL, the term (λ/cp) above and in Table 4.1 is evaluated at the unburned gas condition. Considering that δL has been estimated from graphs, the value for all fuels in the last column of Table 4.1 can certainly be considered of order 1. Figures 4.6–4.8 are the results for the stoichiometric propane–air flame. Figure 4.6 reports the variance of the major species, temperature, and heat release; Figure 4.7 reports the major stable propane fragment distribution due to the proceeding reactions; and Figure 4.8 shows the radical and formaldehyde distributions—all as a function of a spatial distance through the flame wave. As stated, the total wave thickness is chosen from the point at which one of the reactant mole fractions begins to decay to the point at which the heat release rate begins to taper off sharply. Since the point of initial reactant decay corresponds closely to the initial perceptive rise in temperature, the initial thermoneutral period is quite short. The heat release rate curve would ordinarily drop to zero sharply except that the recombination of the radicals in the burned gas zone contribute some energy. The choice of the position that separates the preheat zone and the reaction zone has been made to account for the slight exothermicity of the fuel attack reactions by radicals which have diffused into

174

Combustion

the preheat zone, and the reaction of the resulting species to form water. Note that water and hydrogen exist in the preheat zone. This choice of operation is then made at the point where the heat release rate curve begins to rise sharply. At this point, as well, there is noticeable CO. This certainly establishes the lack of a sharp separation between the preheat and reaction zones discussed earlier in this chapter and indicates that in the case of propane–air flames the zones overlap. On the basis just described, the thickness of the complete propane–air flame wave is about 0.6 mm and the preheat and reaction zones are about 0.3 mm each. Thus, although maximum heat release rate occurs near the maximum flame temperature (if it were not for the radicals recombining), the ignition temperature in the sense of Mallard–Le Chatelier and Zeldovich– Frank-Kamenetskii–Semenov is not very close to the flame temperature. Consistent with the general conditions that occur in flames, the HO2 formed by H atom diffusion upstream maximizes just before the reaction zone. H2O2 would begin to form and decompose to OH radicals. This point is in the 900– 1000 K range known to be the thermal condition for H2O2 decomposition. As would be expected, this point corresponds to the rapid decline of the fuel mole fraction and the onset of radical chain branching. Thus the rapid rise of the radical mole fractions and the formation of the olefins and methane intermediates occur at this point as well (see Figs. 4.7–4.8). The peak of the intermediates is followed by those of formaldehyde, CO, and CO2 in the order described from flow reactor results. Propane disappears well before the end of the reaction zone to form as major intermediates ethene, propene, and methane in magnitudes that the β-scission rule and the type and number of CˆH bonds would have predicted. Likewise, owing to the greater availability of OH radicals after the fuel disappearance, the CO2 concentration begins to rise sharply as the fuel concentration decays. It is not surprising that the depth of the methane–air flame wave is thicker than that of propane–air (Fig. 4.9). Establishing the same criteria for estimating this thickness, the methane–air wave thickness appears to be about 0.9 mm. The thermal thickness is estimated to be 0.5 mm, and the reaction thickness is about 0.4 mm. Much of what was described for the propane–air flame holds for methane–air except as established from the knowledge of methane–air oxidation kinetics; the methane persists through the whole reaction zone and there is a greater overlap of the preheat and reaction zones. Figure 4.10 reveals that at the chosen boundary between the two zones, the methyl radical mole fraction begins to rise sharply. The formaldehyde curve reveals the relatively rapid early conversion of these forms of methyl radicals; that is, as the peroxy route produces ample OH, the methane is more rapidly converted to methyl radical while simultaneously the methyl is converted to formaldehyde. Again, initially, the large mole fraction increases of OH, H, and O is due to H2ˆO2 chain branching at the temperature corresponding to this boundary point. In essence, this point is where explosive reaction occurs and the radical pool is more than sufficient to convert the stable reactants and intermediates to products.

175

Flame Phenomena in Premixed Combustible Gases

If the same criteria are applied to the analysis of the H2–air results in Figs. 4.11–4.12, some initially surprising conclusions are reached. At best, it can be concluded that the flame thickness is approximately 0.5 mm. At most, if any preheat zone exists, it is only 0.1 mm. In essence, then, because of the formation of large H atom concentrations, there is extensive upstream H atom diffusion that causes the sharp rise in HO2. This HO2 reacts with the H2 fuel to form H atoms and H2O2, which immediately dissociates into OH radicals. Furthermore, even at these low temperatures, the OH reacts with the H2 to form water and an abundance of H atoms. This reaction is about 50 kJ exothermic. What appears as a rise in the O2 is indeed only a rise in mole fraction and not in mass. Figure 4.13 reports the results of varying the pressure from 0.5 to 8 atm on the structure of stoichiometric methane–air flames, and Table 4.2 gives the corresponding flame speeds and mass burning rates. Note from Table 4.2 that, as the pressure increases, the flame speed decreases and the mass burning rate increases for the reasons discussed in Section C1. The fact that the temperature profiles in Fig. 4.13 steepen as the pressure rises and that the flame speed results in Table 4.2 decline with pressure would at first appear counterintuitive in light of the simple thermal theories. However, the thermal diffusivity is also pressure dependent and is inversely proportional to the pressure. Thus the thermal diffusivity effect overrides the effect of pressure on the reaction rate and the energy release rate, which affects the temperature distribution. The mass burning rate does increase with pressure although for a very few particular reacting systems either the flame speed or the mass burning rate might not follow

2500

100,000 P  0.25 atm P  1 atm

80,000

Temperature (K)

P  8 atm 1500

60,000

1000

40,000

q  10

500

Heat release rate (J/s cm3)

2000

20,000 q  50

0 0.5

0

0.5

1

1.5

2

0 2.5

Flame coordinate (mm) FIGURE 4.13 Heat release rate and temperature profiles for a stoichiometric CH4–air laminar flame at various pressures and T0  298 K.

176

Combustion

TABLE 4.2 Flame Properties as a Function of Pressurea SL (cm/s)

m 0  ρSL (g/cm2s)

δL [cm (est.)]b

m 0 δL (g/cm s)

0.25

54.51

0.015

0.250

0.0038

1.73

1.00

36.21

0.041

0.085

0.0035

1.59

8.00

18.15

0.163

0.022

0.0036

1.64

P (atm)

a b

( m 0 δL)/(λ/cp)0

CH4–air, φ  1, T0  298 K. Fig. 4.13.

the trends shown. However, for most hydrocarbon–air systems the trends described would hold. As discussed for Table 4.1 and considering that (λ/cp)  f(P), it is not surprising that m 0 δL and (m 0δL )/(λ /c p )0 in Table 4.2 essentially do not vary with pressure and remain of order 1.

4. The Laminar Flame and the Energy Equation An important point about laminar flame propagation—one that has not previously been discussed—is worth stressing. It has become common to accept that reaction rate phenomena dominate in premixed homogeneous combustible gaseous mixtures and diffusion phenomena dominate in initially unmixed fuel–oxidizer systems. (The subject of diffusion flames will be discussed in Chapter 6.) In the case of laminar flames, and indeed in most aspects of turbulent flame propagation, it should be emphasized that it is the diffusion of heat (and mass) that causes the flame to propagate; that is, flame propagation is a diffusional mechanism. The reaction rate determines the thickness of the reaction zone and, thus, the temperature gradient. The temperature effect is indeed a strong one, but flame propagation is still attributable to the diffusion of heat and mass. The expression SL  (αRR)1/2 says it well—the propagation rate is proportional to the square root of the diffusivity and the reaction rate.

5. Flame Speed Measurements For a long time there was no interest in flame speed measurements. Sufficient data and understanding were thought to be at hand. But as lean burn conditions became popular in spark ignition engines, the flame speed of lean limits became important. Thus, interest has been rekindled in measurement techniques. Flame velocity has been defined as the velocity at which the unburned gases move through the combustion wave in a direction normal to the wave surface.

Flame Phenomena in Premixed Combustible Gases

ur

177

ur

ux T1

ux T1

r,x FIGURE 4.14 Velocity and temperature variations through non-one-dimensional flame systems.

If, in an infinite plane flame, the flame is regarded as stationary and a particular flow tube of gas is considered, the area of the flame enclosed by the tube does not depend on how the term “flame surface or wave surface” in which the area is measured is defined. The areas of all parallel surfaces are the same, whatever property (particularly temperature) is chosen to define the surface; and these areas are all equal to each other and to that of the inner surface of the luminous part of the flame. The definition is more difficult in any other geometric system. Consider, for example, an experiment in which gas is supplied at the center of a sphere and flows radially outward in a laminar manner to a stationary spherical flame. The inward movement of the flame is balanced by the outward flow of gas. The experiment takes place in an infinite volume at constant pressure. The area of the surface of the wave will depend on where the surface is located. The area of the sphere for which T  500°C will be less than that of one for which T  1500°C. So if the burning velocity is defined as the volume of unburned gas consumed per second divided by the surface area of the flame, the result obtained will depend on the particular surface selected. The only quantity that does remain constant in this system is the product urρrAr, where ur is the velocity of flow at the radius r, where the surface area is Ar, and the gas density is ρr. This product equals m r , the mass flowing through the layer at r per unit time, and must be constant for all values of r. Thus, ur varies with r the distance from the center in the manner shown in Fig. 4.14. It is apparent from Fig. 4.14 that it is difficult to select a particular linear flow rate of unburned gas up to the flame and regard this velocity as the burning velocity. If an attempt is made to define burning velocity strictly for such a system, it is found that no definition free from all possible objections can be formulated. Moreover, it is impossible to construct a definition that will, of necessity,

178

Combustion

determine the same value as that found in an experiment using a plane flame. The essential difficulties are as follow: (1) over no range of r values does the linear velocity of the gas have even an approximately constant value and (2) in this ideal system, the temperature varies continuously from the center of the sphere outward and approaches the flame surface asymptotically as r approaches infinity. So no spherical surface can be considered to have a significance greater than any other. In Fig. 4.14, ux, the velocity of gas flow at x for a plane flame, is plotted on the same scale against x, the space coordinate measured normal to the flame front. It is assumed that over the main part of the rapid temperature rise, ur and ux coincide. This correspondence is likely to be true if the curvature of the flame is large compared with the flame thickness. The burning velocity is then, strictly speaking, the value to which ux approaches asymptotically as x approaches . However, because the temperature of the unburned gas varies exponentially with x, the value of ux becomes effectively constant only a very short distance from the flame. The value of ur on the low-temperature side of the spherical flame will not at any point be as small as the limiting value of ux. In fact, the difference, although not zero, will probably not be negligible for such flames. This value of ur could be determined using the formula ur  m /ρr Ar Since the layer of interest is immediately on the unburned side of the flame,  ρr will be close to ρu, the density of the unburned gas, and m/ρ will be close to the volume flow rate of unburned gas. So, to obtain, in practice, a value for burning velocity close to that for the plane flame, it is necessary to locate and measure an area as far on the unburned side of the flame as possible. Systems such as Bunsen flames are in many ways more complicated than either the plane case or the spherical case. Before proceeding, consider the methods of observation. The following methods have been most widely used to observe the flame: (a) The luminous part of the flame is observed, and the side of this zone, which is toward the unburned gas, is used for measurement (direct photograph). (b) A shadowgraph picture is taken. (c) A Schlieren picture is taken. (d) Interferometry (a less frequently used method). Which surface in the flame does each method give? Again consider the temperature distribution through the flame as given in Fig. 4.15. The luminous zone comes late in the flame and thus is generally not satisfactory. A shadowgraph picture measures the derivative of the density gradient (∂ρ/∂x) or (1/T2)(∂T/∂x); that is, it evaluates {∂[(1/T2)(∂T/∂x)]/∂x}  (2/T3)

179

Flame Phenomena in Premixed Combustible Gases

T Tb  Tf Preheat zone

Ti Reaction zone Luminous zone

Tu  To

x FIGURE 4.15 Temperature regimes in a laminar flame. Visible edge Schlieren edge Inner shadow cone

FIGURE 4.16 Optical fronts in a Bunsen burner flame.

(∂T/∂x)2  (1/T2)(∂2T/∂x2). Shadowgraphs, therefore measure the earliest variational front and do not precisely specify a surface. Actually, it is possible to define two shadowgraph surfaces—one at the unburned side and one on the burned side. The inner cone is much brighter than the outer cone, since the absolute value for the expression above is greater when evaluated at T0 than at Tf. Schlieren photography gives simply the density gradient (∂ρ/∂x) or (1/T2) (∂T/∂x), which has the greatest value about the inflection point of the temperature curve; it also corresponds more closely to the ignition temperature. This surface lies quite early in the flame, is more readily definable than most images, and is recommended and preferred by many workers. Interferometry, which measures density or temperature directly, is much too sensitive and can be used only on two-dimensional flames. In an exaggerated picture of a Bunsen tube flame, the surfaces would lie as shown in Fig. 4.16. The various experimental configurations used for flame speeds may be classified under the following headings: (a) Conical stationary flames on cylindrical tubes and nozzles (b) Flames in tubes

180

Combustion

(c) Soap bubble method (d) Constant volume explosion in spherical vessel (e) Flat flame methods. The methods are listed in order of decreasing complexity of flame surface and correspond to an increasing complexity of experimental arrangement. Each has certain advantages that attend its usage.

a. Burner Method In this method premixed gases flow up a jacketed cylindrical tube long enough to ensure streamline flow at the mouth. The gas burns at the mouth of the tube, and the shape of the Bunsen cone is recorded and measured by various means and in various ways. When shaped nozzles are used instead of long tubes, the flow is uniform instead of parabolic and the cone has straight edges. Because of the complicated flame surface, the different procedures used for measuring the flame cone have led to different results. The burning velocity is not constant over the cone. The velocity near the tube wall is lower because of cooling by the walls. Thus, there are lower temperatures, which lead to lower reaction rates and, consequently, lower flame speeds. The top of the cone is crowded owing to the large energy release; therefore, reaction rates are too high. It has been found that 30% of the internal portion of the cone gives a constant flame speed when related to the proper velocity vector, thereby giving results comparable with other methods. Actually, if one measures SL at each point, one will see that it varies along every point for each velocity vector, so it is not really constant. This variation is the major disadvantage of this method. The earliest procedure of calculating flame speed was to divide the volume flow rate (cm3 s1) by the area (cm2) of flame cone: SL 

Q cm 3 s1  cm s1 A cm 2

It is apparent, then, that the choice of cone surface area will give widely different results. Experiments in which fine magnesium oxide particles are dispersed in the gas stream have shown that the flow streamlines remain relatively unaffected until the Schlieren cone, then diverge from the burner axis before reaching the visible cone. These experiments have led many investigators to use the Schlieren cone as the most suitable one for flame speed evaluation. The shadow cone is used by many experimenters because it is much simpler than the Schlieren techniques. Moreover, because the shadow is on the cooler side, it certainly gives more correct results than the visible cone. However, the flame cone can act as a lens in shadow measurements, causing uncertainties to arise with respect to the proper cone size.

Flame Phenomena in Premixed Combustible Gases

181

SL α α uu

FIGURE 4.17 Velocity vectors in a Bunsen core flame.

Some investigators have concentrated on the central portion of the cone only, focusing on the volume flow through tube radii corresponding to this portion. The proper choice of cone is of concern here also. The angle the cone slant makes with the burner axis can also be used to determine SL (see Fig. 4.17). This angle should be measured only at the central portion of the cone. Thus SL  uu sin α. Two of the disadvantages of the burner methods are 1. Wall effects can never be completely eliminated. 2. A steady source of gas supply is necessary, which is hard to come by for rare or pure gases. The next three methods to be discussed make use of small amounts of gas.

b. Cylindrical Tube Method In this method, a gas mixture is placed in a horizontal tube opened at one end; then the mixture is ignited at the open end of the tube. The rate of progress of the flame into the unburned gas is the flame speed. The difficulty with this method is that, owing to buoyancy effects, the flame front is curved. Then the question arises as to which flame area to use. The flame area is no longer a geometric image of the tube; if it is hemispherical, SLAf  umπR2. Closer observation also reveals quenching at the wall. Therefore, the unaffected center mixes with an affected peripheral area. Because a pressure wave is established by the burning (recall that heating causes pressure change), the statement that the gas ahead of the flame is not affected by the flame is incorrect. This pressure wave causes a velocity in the unburned gases, so one must account for this movement. Therefore, since the flame is in a moving gas, this velocity must be subtracted from the measured value. Moreover, friction effects downstream generate a greater pressure wave; therefore, length can have an effect. One can deal with this by capping the end of the tube, drilling a small hole in the cap, and measuring the efflux with a

182

Combustion

soap solution [18]. The rate of growth of the resultant soap bubble is used to obtain the velocity exiting the tube, and hence the velocity of unburned gas. A restriction at the open end minimizes effects due to the back flow of the expanding burned gases. These adjustments permit relatively good values to be obtained, but still there are errors due to wall effects and distortion due to buoyancy. This buoyancy effect can be remedied by turning the tube vertically.

c. Soap Bubble Method In an effort to eliminate wall effects, two spherical methods were developed. In the one discussed here, the gas mixture is contained in a soap bubble and ignited at the center by a spark so that a spherical flame spreads radially through the mixture. Because the gas is enclosed in a soap film, the pressure remains constant. The growth of the flame front along a radius is followed by some photographic means. Because, at any stage of the explosion, the burned gas behind the flame occupies a larger volume than it did as unburned gas, the fresh gas into which the flame is burning moves outward. Then SL Aρ0  ur Aρf ⎛ Amount of material ⎞⎟ ⎜⎜ ⎟ ⎜⎜ that must go into ⎟⎟ ⎜⎜ flame to increase ⎟⎟⎟  velocity observed ⎟⎟ ⎜⎜ ⎟⎠ ⎜⎝ volume SL  ur (ρf /ρ0 ) The great disadvantage is the large uncertainty in the temperature ratio T0/Tf necessary to obtain ρf/ρ0. Other disadvantages are the facts that (1) the method can only be used for fast flames to avoid the convective effect of hot gases and (2) the method cannot work with dry mixtures.

d. Closed Spherical Bomb Method The bomb method is quite similar to the bubble method except that the constant volume condition causes a variation in pressure. One must, therefore, follow the pressure simultaneously with the flame front. As in the soap bubble method, only fast flames can be used because the adiabatic compression of the unburned gases must be measured in order to calculate the flame speed. Also, the gas into which the flame is moving is always changing; consequently, both the burning velocity and flame speed vary throughout the explosion. These features make the treatment complicated and, to a considerable extent, uncertain. The following expression has been derived for the flame speed [19]: ⎡ R3  r 3 dP ⎤⎥ dr SL  ⎢⎢1  3P γ u r 2 dr ⎥⎦ dt ⎣

183

Flame Phenomena in Premixed Combustible Gases

where R is the sphere radius and r is the radius of spherical flames at any moment. The fact that the second term in the brackets is close to 1 makes it difficult to attain high accuracy.

e. Flat Flame Burner Method The flame burner method is usually attributed to Powling [20]. Because it offers the simplest flame front—one in which the area of shadow, Schlieren, and visible fronts are all the same—it is probably the most accurate. By placing either a porous metal disk or a series of small tubes (1 mm or less in diameter) at the exit of the larger flow tube, one can create suitable conditions for flat flames. The flame is usually ignited with a high flow rate, then the flow or composition is adjusted until the flame is flat. Next, the diameter of the flame is measured, and the area is divided into the volume flow rate of unburned gas. If the velocity emerging is greater than the flame speed, one obtains a cone due to the larger flame required. If velocity is too slow, the flame tends to flash back and is quenched. In order to accurately define the edges of the flame, an inert gas is usually flowed around the burners. By controlling the rate of efflux of burned gases with a grid, a more stable flame is obtained. This experimental apparatus is illustrated in Fig. 4.18. As originally developed by Powling, this method was applicable only to mixtures having low burning velocities of the order of 15 cm/s and less. At higher burning velocities, the flame front positions itself too far from the burner and takes a multiconical form. Later, however, Spalding and Botha [21] extended the flat flame burner method to higher flame speeds by cooling the plug. The cooling draws the flame front closer to the plug and stabilizes it. Operationally, the procedure is as follows. A flow rate giving a velocity greater than the flame speed is set, and the cooling is controlled until a flat flame is obtained. For a given mixture ratio many cooling rates are used. A plot of flame speed versus cooling rate is made and extrapolated to zero-cooling rate (Fig. 4.19). At this point the adiabatic flame speed SL is obtained. This procedure can be used for all mixture ratios within the

Grid Flame 

Inert gas

Gas mixture

FIGURE 4.18 Flat flame burner apparatus.

184

Combustion

Adiabatic

SL (At constant Tu calculation)

0 FIGURE 4.19

q Cooling rate

Cooling effect in flat flame burner apparatus.

flammability limits. This procedure is superior to the other methods because the heat that is generated leaks to the porous plug, not to the unburned gases as in the other model. Thus, quenching occurs all along the plug, not just at the walls. The temperature at which the flame speed is measured is calculated as follows. For the approach gas temperature, one calculates what the initial temperature would have been if there were no heat extraction. Then the velocity of the mixture, which would give the measured mass flow rate at this temperature, is determined. This velocity is SL at the calculated temperature. Detailed descriptions of various burned systems and techniques are to be found in Ref. [22]. A similar flat flame technique—one that does not require a heat loss correction—is the so-called opposed-jet system. This approach to measuring flame speeds was introduced to determine the effect of flame stretch on the measured laminar flame velocity. The concept of stretch was introduced in attempts to understand the effects of turbulence on the mass burning rate of premixed systems. (This subject is considered in more detail in Section E.) The technique uses two opposing jets of the same air–fuel ratio to create an almost planar stagnation plane with two flat flames on both sides of the plane. For the same mixture ratio, stable flames are created for different jet velocities. In each case, the opposing jets have the same exit velocity. The velocity leaving a jet decreases from the jet exit toward the stagnation plane. This velocity gradient is related to the stretch affecting the flames: the larger the gradient, the greater the stretch. Measurements are made for different gradients for a fixed mixture. A plot is then made of the flame velocity as a function of the calculated stress function (velocity gradient), and the values are extrapolated to zero velocity gradient. The extrapolated value is considered to be the flame velocity free from any stretch effects—a value that can be compared to theoretical calculations that do not account for the stretch factor. The same technique is used to evaluate diffusion flames in which one jet contains the fuel

Flame Phenomena in Premixed Combustible Gases

185

and the other the oxidizer. Figures depicting opposed-jet systems are shown in Chapter 6. The effect of stretch on laminar premixed flame speeds is generally slight for most fuels in air.

6. Experimental Results: Physical and Chemical Effects The Mallard–Le Chatelier development for the laminar flame speed permits one to determine the general trends with pressure and temperature. When an overall rate expression is used to approximate real hydrocarbon oxidation kinetics experimental results, the activation energy of the overall process is found to be quite high—of the order of 160 kJ/mol. Thus, the exponential in the flame speed equation is quite sensitive to variations in the flame temperature. This sensitivity is the dominant temperature effect on flame speed. There is also, of course, an effect of temperature on the diffusivity; generally, the diffusivity is considered to vary with the temperature to the 1.75 power. The pressure dependence of flame speed as developed from the thermal approaches was given by the expression SL ∼ [ P (n2 ) ]1/ 2

(4.22)

where n was the overall order of reaction. Thus, for second-order reactions the flame speed appears independent of pressure. In observing experimental measurements of flame speed as a function of pressure, one must determine whether the temperature was kept constant with inert dilution. As the pressure is increased, dissociation decreases and the temperature rises. This effect must be considered in the experiment. For hydrocarbon–air systems, however, the temperature varies little from atmospheric pressure and above due to a minimal amount of dissociation. There is a more pronounced temperature effect at subatmospheric pressures. To a first approximation one could perhaps assume that hydrocarbon–air reactions are second-order. Although it is impossible to develop a single overall rate expression for the complete experimental range of temperatures and pressures used by various investigators, values have been reported and hold for the limited experimental ranges of temperature and pressure from which the expression was derived. The overall reaction orders reported range from 1.5 to 2.0, and most results are around 1.75 [2, 23]. Thus, it is not surprising that experimental results show a decline in flame speed with increasing pressure [2]. As briefly mentioned earlier, with the background developed in the detailed studies of hydrocarbon oxidation, it is possible to explain this pressure trend more thoroughly. Recall that the key chain branching reaction in any hydrogencontaining system is the following reaction (3.15): H  O2 → O  OH

(4.62)

186

Combustion

Any process that reduces the H atom concentration and any reaction that competes with reaction (4.62) for H atoms will tend to reduce the overall oxidation rate; that is, it will inhibit combustion. As discussed in reaction (3.21), reaction (4.63) H  O2  M → HO2  M

(4.63)

Laminar burning velocity (cm/s)

competes directly with reaction (4.62). Reaction (4.63) is third-order and therefore much more pressure dependent than reaction (4.62). Consequently, as pressure is increased, reaction (4.63) essentially inhibits the overall reaction and reduces the flame speed. Figure 4.20 reports the results of some analytical calculations of flame speeds in which detailed kinetics were included; the results obtained are quite consistent with recent measurements [2]. For pressures below atmospheric, there is only a very small decrease in flame speed as the pressure is increased; and at higher pressure (1–5 atm), the decline in SL with increasing pressure becomes more pronounced. The reason for this change of behavior is twofold. Below atmospheric pressure, reaction (4.63) does not compete effectively with reaction (4.62) and any decrease due to reaction (4.63) is compensated by a rise in temperature. Above 1 atm reaction (4.63) competes very effectively with reaction (4.62); temperature variation with pressure in this range is slight, and thus a steeper decline in SL with pressure is found. Since the kinetic and temperature trends with pressure exist for all hydrocarbons, the same pressure effect on SL will exist for all such fuels. Even though SL decreases with increasing pressure for the conditions described, m 0 increases with increasing pressure because of the effect of pressure on ρ0. And for higher O2 concentrations, the temperature rises substantially,

100 80 60

C2H4 CH3OH

40

CH4

20

0.1

1

10

Pressure (atm) FIGURE 4.20 Variation in laminar flame speeds with pressure for some stoichiometric fuel–air mixtures (after Westbrook and Dryer [2]).

187

Flame Phenomena in Premixed Combustible Gases

about 30% for pure O2; thus the point where reaction (4.63) can affect the chain branching step reaction (4.62) goes to much higher pressure. Consequently, in oxygen-rich systems SL almost always increases with pressure. The variation of flame speed with equivalence ratio follows the variation with temperature. Since flame temperatures for hydrocarbon–air systems peak slightly on the fuel-rich side of stoichiometric (as discussed in Chapter 1), so do the flame speeds. In the case of hydrogen–air systems, the maximum SL falls well on the fuel-rich side of stoichiometric, since excess hydrogen increases the thermal diffusivity substantially. Hydrogen gas with a maximum value of 325 cm/s has the highest flame speed in air of any other fuel. Reported flame speed results for most fuels vary somewhat with the measurement technique used. Most results, however, are internally consistent. Plotted in Fig. 4.21 are some typical flame speed results as a function of the stoichiometric mixture ratio. Detailed data, which were given in recent combustion symposia, are available in the extensive tabulations of Refs. [24–26]. The flame speeds for many fuels in air have been summarized from these references and are listed in Appendix F. Since most paraffins, except methane, have approximately the same flame temperature in air, it is not surprising that their flame speeds are about the same (45 cm/s). Methane has a somewhat lower speed ( 40 cm/s). Attempts [24] have been made to correlate flame speed with hydrocarbon fuel structure and chain length, but these correlations

400

H2

SL (cm/s)

300

200 C2H2 100 80 60

C6H6

40 20 0

C2H4 C3H8

CO

CH4 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 φ

FIGURE 4.21 General variation in laminar flame speeds with equivalence ratio φ for various fuel–air systems at P  1 atm and T0  298 K.

188

Combustion

Flame velocity relative to ψ  0.21 SL rel

appear to follow the general trends of temperature. Olefins, having the same C/H ratio, have the same flame temperature (except for ethene, which is slightly higher) and have flame speeds of approximately 50 cm/s. In this context ethene has a flame speed of approximately 75 cm/s. Owing to its high flame temperature, acetylene has a maximum flame speed of about 160 cm/s. Molecular hydrogen peaks far into the fuel-rich region because of the benefit of the fuel diffusivity. Carbon monoxide favors the rich side because the termination reaction H  CO  M → HCO  M is a much slower step than the termination step H  O2  M → HO2  M, which would prevail in the lean region. The variation of flame speed with oxygen concentration poses further questions about the factors that govern the flame speed. Shown in Fig. 4.22 is the flame speed of a fuel in various oxygen–nitrogen mixtures relative to its value in air. Note the 10-fold increase for methane between pure oxygen and air, the 7.5-fold increase for propane, the 3.4-fold increase for hydrogen, and the 2.4-fold increase for carbon monoxide. From the effect of temperature on the overall rates and diffusivities, one would expect about a fivefold increase for all these fuels. Since the CO results contain a fixed amount of hydrogen additives [24], the fact that the important OH radical concentration does not increase as much as expected must play a role in the lower rise. Perhaps for general considerations the hydrogen values are near enough to a general estimate. Indeed, there is probably a sufficient radical pool at all oxygen concentrations. For the hydrocarbons, the radical pool concentration undoubtedly increases substantially as one goes to pure oxygen for two reasons—increased temperature and no nitrogen dilution. Thus, applying the same general rate expression for air and oxygen just does not suffice. 1200

Methane Methane Methane Methane Propane Ethane Acetylene Hydrogen Carbon Monoxide

600

0

0

0.2 0.4 0.6 0.8 Mole fraction of oxygen, ψ O2 O2  N2

1.0

FIGURE 4.22 Relative effect of oxygen concentrations on flame speed for various fuel–air systems at P  1 atm and T0  298 K (after Zebatakis [25]).

189

Flame Phenomena in Premixed Combustible Gases

The effect of the initial temperature of a premixed fuel–air mixture on the flame propagation rate again appears to be reflected through the final flame temperature. Since the chemical energy release is always so much greater than the sensible energy of the reactants, small changes of initial temperature generally have little effect on the flame temperature. Nevertheless, the flame propagation expression contains the flame temperature in an exponential term; thus, as discussed many times previously, small changes in flame temperature can give noticeable changes in flame propagation rates. If the initial temperatures are substantially higher than normal ambient, the rate of reaction (4.63) can be reduced in the preheat zone. Since reaction (4.63) is one of recombination, its rate decreases with increasing temperature, and so the flame speed will be attenuated even further. Perhaps the most interesting set of experiments to elucidate the dominant factors in flame propagation was performed by Clingman et al. [27]. Their results clearly show the effect of the thermal diffusivity and reaction rate terms. These investigators measured the flame propagation rate of methane in various oxygen–inert gas mixtures. The mixtures of oxygen to inert gas were 0.21/0.79 on a volumetric basis, the same as that which exists for air. The inerts chosen were nitrogen (N2), helium (He), and argon (Ar). The results of these experiments are shown in Fig. 4.23. The trends of the results in Fig. 4.23 can be readily explained. Argon and nitrogen have thermal diffusivities that are approximately equal. However, Ar is a monatomic gas whose specific heat is lower than that of N2. Since the

140

Burning velocity (cm/s)

120

He

100 80

Ar

60 N2

40 20 0

4

6

8

10

12

14

16

CH4 in various airs (%) FIGURE 4.23 Methane laminar flame velocities in various inert gas–oxygen mixtures (after Clingman et al. [27]).

190

Combustion

heat release in all systems is the same, the final (or flame) temperature will be higher in the Ar mixture than in the N2 mixture. Thus, SL will be higher for Ar than for N2. Argon and helium are both monatomic, so their final temperatures are equal. However, the thermal diffusivity of He is much greater than that of Ar. Helium has a higher thermal conductivity and a much lower density than argon. Consequently, SL for He is much greater than that for Ar. The effect of chemical additives on the flame speed has also been explored extensively. Leason [28] has reported the effects on flame velocity of small concentrations of additive ( 3%) and other fuels. He studied the propane–air flame. Among the compounds considered were acetone, acetaldehyde, benzaldehyde, diethyl ether, benzene, and carbon disulfide. In addition, many others were chosen from those classes of compounds that were shown to be oxidation intermediates in low-temperature studies; these compounds were expected to decrease the induction period and, thus, increase the flame velocity. Despite differences in apparent oxidation properties, all the compounds studied changed the flame velocity in exactly the same way that dilution with excess fuel would on the basis of oxygen requirement. These results support the contention that the laminar flame speed is controlled by the high-temperature reaction region. The high temperatures generate more than ample radicals via chain branching, so it is unlikely that any additive could contribute any reaction rate accelerating feature. There is, of course, a chemical effect in carbon monoxide flames. This point was mentioned in the discussion of carbon monoxide explosion limits. Studies have shown that CO flame velocities increase appreciably when small amounts of hydrogen, hydrogen-containing fuels, or water are added. For 45% CO in air, the flame velocity passes through a maximum after approximately 5% by volume of water has been added. At this point, the flame velocity is 2.1 times the value with 0.7% H2O added. After the 5% maximum is attained a dilution effect begins to cause a decrease in flame speed. The effect and the maximum arise because a sufficient steady-state concentration of OH radicals must be established for the most effective explosive condition. Although it may be expected that the common antiknock compounds would decrease the flame speed, no effects of antiknocks have been found in constant pressure combustion. The effect of the inhibition of the preignition reaction on flame speed is of negligible consequence. There is no universal agreement on the mechanism of antiknocks, but it has been suggested that they serve to decrease the radical concentrations by absorption on particle surfaces (see Chapter 2). The reduction of the radical concentration in the preignition reactions or near the flammability limits can severely affect the ability to initiate combustion. In these cases the radical concentrations are such that the chain branching factor is very close to the critical value for explosion. Any reduction could prevent the explosive condition from being reached. Around the stoichiometric mixture ratio, the radical concentrations are normally so great that it is most difficult to add any small amounts of additives that would capture enough radicals to alter the reaction rate and the flame speed.

Flame Phenomena in Premixed Combustible Gases

191

Certain halogen compounds, such as the Freons, are known to alter the flammability limits of hydrocarbon–air mixtures. The accepted mechanism is that the halogen atoms trap hydrogen radicals necessary for the chain branching step. Near the flammability limits, conditions exist in which the radical concentrations are such that the chain branching factor α is just above αcrit. Any reduction in radicals and the chain branching effects these radicals engender could eliminate the explosive (fast reaction rate and larger energy release rate) regime. However, small amounts of halogen compounds do not seem to affect the flame speed in a large region around the stoichiometric mixture ratio. The reason is, again, that in this region the temperatures are so high and radicals so abundant that elimination of some radicals does not affect the reaction rate. It has been found that some of the larger halons (the generic name for the halogenated compounds sold under commercial names such as Freon) are effective flame suppressants. Also, some investigators have found that inert powders are effective in fire fighting. Fundamental experiments to evaluate the effectiveness of the halons and powders have been performed with various types of apparatus that measure the laminar flame speed. Results have indicated that the halons and the powders reduce flame speeds even around the stoichiometric air–fuel ratio. The investigators performing these experiments have argued that those agents are effective because they reduce the radical concentrations. However, this explanation could be questioned. The quantities of these added agents are great enough that they could absorb sufficient amounts of heat to reduce the temperature and hence the flame speed. Both halons and powders have large total heat capacities.

D. STABILITY LIMITS OF LAMINAR FLAMES There are two types of stability criteria associated with laminar flames. The first is concerned with the ability of the combustible fuel–oxidizer mixture to support flame propagation and is strongly related to the chemical rates in the system. In this case a point can be reached for a given limit mixture ratio in which the rate of reaction and its subsequent heat release are not sufficient to sustain reaction and, thus, propagation. This type of stability limit includes (1) flammability limits in which gas-phase losses of heat from limit mixtures reduce the temperature, rate of heat release, and the heat feedback, so that the flame is not permitted to propagate and (2) quenching distances in which the loss of heat to a wall and radical quenching at the wall reduce the reaction rate so that it cannot sustain a flame in a confined situation such as propagation in a tube. The other type of stability limit is associated with the mixture flow and its relationship to the laminar flame itself. This stability limit, which includes the phenomena of flashback, blowoff, and the onset of turbulence, describes the limitations of stabilizing a laminar flame in a real experimental situation.

192

Combustion

1. Flammability Limits The explosion limit curves presented earlier and most of those that appear in the open literature are for a definite fuel–oxidizer mixture ratio, usually stoichiometric. For the stoichiometric case, if an ignition source is introduced into the mixture even at a very low temperature and at reasonable pressures (e.g., 1 atm), the gases about the ignition source reach a sufficient temperature so that the local mixture moves into the explosive region and a flame propagates. This flame, of course, continues to propagate even after the ignition source is removed. There are mixture ratios, however, that will not self-support the flame after the ignition source is removed. These mixture ratios fall at the lean and rich end of the concentration spectrum. The leanest and richest concentrations that will just self-support a flame are called the lean and rich flammability limits, respectively. The primary factor that determines the flammability limit is the competition between the rate of heat generation, which is controlled by the rate of reaction and the heat of reaction for the limit mixture, and the external rate of heat loss by the flame. The literature reports flammability limits in both air and oxygen. The lean limit rarely differs for air or oxygen, as the excess oxygen in the lean condition has the same thermophysical properties as nitrogen. Some attempts to standardize the determination of flammability limits have been made. Coward and Jones [29] recommended that a 2-in. glass tube about 4 ft long be employed; such a tube should be ignited by a spark a few millimeters long or by a small naked flame. The high-energy starting conditions are such that weak mixtures will be sure to ignite. The large tube diameter is selected because it gives the most consistent results. Quenching effects may interfere in tubes of small diameter. Large diameters create some disadvantages since the quantity of gas is a hazard and the possibility of cool flames exists. The 4-foot length is chosen in order to allow an observer to truly judge whether the flame will propagate indefinitely or not. It is important to specify the direction of flame propagation. Since it may be assumed as an approximation that a flame cannot propagate downward in a mixture contained within a vertical tube if the convection current it produces is faster than the speed of the flame, the limits for upward propagation are usually slightly wider than those for downward propagation or those for which the containing tube is in a horizontal position. Table 4.3 lists some upper and lower flammability limits (in air) taken from Refs. [24] and [25] for some typical combustible compounds. Data for other fuels are given in Appendix F. In view of the accelerating effect of temperature on chemical reactions, it is reasonable to expect that limits of flammability should be broadened if the temperature is increased. This trend is confirmed experimentally. The increase is slight and it appears to give a linear variation for hydrocarbons. As noted from the data in Appendix E, the upper limit for many fuels is about 3 times stoichiometric and the lower limit is about 50% of stoichiometric.

193

Flame Phenomena in Premixed Combustible Gases

TABLE 4.3 Flammability Limits of Some Fuels in Aira Lower (lean)

Upper (rich)

Methane

5

15

Heptane

1

Hydrogen

4

Carbon monoxide

Stoichiometric 9.47

6.7

12.5

1.87

75

29.2

74.2

29.5

Acetaldehyde

4.0

60

7.7

Acetylene

2.5

100

7.7

Carbon disulfide

1.3

50

7.7

Ethylene oxide

3.6

100

7.7

a

Volume percent.

TABLE 4.4 Comparison of Oxygen and Air Flammability Limitsa Lean

Rich

Air

O2

Air

O2

H2

4

4

75

94

CO

12

16

74

94

NH3

15

15

28

79

CH4

5

5

15

61

C3H8

2

2

10

55

a

Fuel volume percent.

Generally, the upper limit is higher than that for detonation. The lower (lean) limit of a gas is the same in oxygen as in air owing to the fact that the excess oxygen has the same heat capacity as nitrogen. The higher (rich) limit of all flammable gases is much greater in oxygen than in air, due to higher temperature, which comes about from the absence of any nitrogen. Hence, the range of flammability is always greater in oxygen. Table 4.4 shows this effect. As increasing amounts of an incombustible gas or vapor are added to the atmosphere, the flammability limits of a gaseous fuel in the atmosphere approach one another and finally meet. Inert diluents such as CO2, N2, or Ar merely replace part of the O2 in the mixture, but these inert gases do not have

194

Combustion

the same extinction power. It is found that the order of efficiency is the same as that of the heat capacities of these three gases: CO2 N 2 Ar (or He) For example, the minimum percentage of oxygen that will permit flame propagation in mixtures of CH4, O2, and CO2 is 14.6%; if N2 is the diluent, the minimum percentage of oxygen is less and equals 12.1%. In the case of Ar, the value is 9.8%. As discussed, when a gas of higher specific heat is present in sufficient quantities, it will reduce the final temperature, thereby reducing the rate of energy release that must sustain the rate of propagation over other losses. It is interesting to examine in more detail the effect of additives as shown in Fig. 4.24 [25]. As discussed, the general effect of the nonhalogenated additives follows the variation in the molar specific heat; that is, the greater the specific heat of an inert additive, the greater the effectiveness. Again, this effect is strictly one of lowering the temperature to a greater extent; this was verified as well by flammability measurements in air where the nitrogen was replaced by carbon dioxide and argon. Figure 4.24, however, represents the condition in which additional species were added to the air in the fuel–air mixture. As noted in Fig. 4.24, 16

14

Methane volume (%)

12 Flammable mixtures 10

8

MeBr

CCL4 CO2 H2O

6

N2 He

4

% air  100%  % CH4  % inert

2

0

0

20 30 40 10 Added inert volume (%)

50

FIGURE 4.24 Limits of flammability of various methane–inert gas–air mixtures at P  1 and T0  298 K (after Zebatakis [25]).

Flame Phenomena in Premixed Combustible Gases

195

rich limits are more sensitive to inert diluents than lean limits; however, species such as halogenated compounds affect both limits and this effect is greater than that expected from heat capacity alone. Helium addition extends the lean limit somewhat because it increases the thermal diffusivity and, thus, the flame speed. That additives affect the rich limit more than the lean limit can be explained by the important competing steps for possible chain branching. When the system is rich [reaction (3.23)], H  H  M → H2  M

(4.64)

competes with [reaction (3.15)] H  O2 → OH  O

(4.62)

The recombination [reaction (4.64)] increases with decreasing temperature and increasing concentration of the third body M. Thus, the more diluent added, the faster this reaction is compared to the chain branching step [reaction (4.62)]. This aspect is also reflected in the overall activation energy found for rich systems compared to lean systems. Rich systems have a much higher overall activation energy and therefore a greater temperature sensitivity. The effect of all halogen compounds on flammability limits is substantial. The addition of only a few percent can make some systems nonflammable. These observed results support the premise that the effect of halogen additions is not one of dilution alone, but rather one in which the halogens act as catalysts in reducing the H atom concentration necessary for the chain branching reaction sequence. Any halogen added—whether in elemental form, as hydrogen halide, or bound in an organic compound—will act in the same manner. Halogenated hydrocarbons have weak carbon–halogen bonds that are readily broken in flames of any respectable temperature, thereby placing a halogen atom in the reacting system. This halogen atom rapidly abstracts a hydrogen from the hydrocarbon fuel to form the hydrogen halide; then the following reaction system, in which X represents any of the halogens F, Cl, Br, or I, occurs: HX  H → H 2  X

(4.65)

X  X  M → X2  M

(4.66)

X 2  H → HX  X

(4.67)

Reactions (4.65)–(4.67) total overall to H  H → H2 and thus it is seen that X is a homogeneous catalyst for recombination of the important H atoms. What is significant in the present context is that the halide

196

Combustion

reactions above are fast compared to the other important H atom reactions such as H  O2 → O  OH

(4.62)

H  RH → R  H 2

(4.68)

or

This competition for H atoms reduces the rate of chain branching in the important H  O2 reaction. The real key to this type of inhibition is the regeneration of X2, which permits the entire cycle to be catalytic. Because sulfur dioxide (SO2) essentially removes O atoms catalytically by the mechanism SO2  O  M → SO3  M

(4.69)

SO3  O → SO2  O2

(4.70)

and also by H radical removal by the system SO2  H   → HSO2  M

(4.71)

HSO2   → SO2  H 2 O

(4.72)

SO2  O  M → SO3  M

(4.73)

SO3  H  M → HSO3  M

(4.74)

HSO3  H → SO2  H 2 O

(4.75)

and by

SO2 is similarly a known inhibitor that affects flammability limits. These catalytic cycles [reactions (4.69)–(4.70), reactions (4.71)–(4.72), and reactions (4.73)–(4.75)] are equivalent to O  O → O2 H  OH → H 2 O H  H  O → H2 O The behavior of flammability limits at elevated pressures can be explained somewhat satisfactorily. For simple hydrocarbons (ethane, propane,…, pentane),

197

Flame Phenomena in Premixed Combustible Gases

it appears that the rich limits extend almost linearly with increasing pressure at a rate of about 0.13 vol%/atm; the lean limits, on the other hand, are at first extended slightly and thereafter narrowed as pressure is increased to 6 atm. In all, the lean limit appears not to be affected appreciably by the pressure. Figure 4.25 for natural gas in air shows the pressure effect for conditions above atmospheric. Most early studies of flammability limits at reduced pressures indicated that the rich and lean limits converge as the pressure is reduced until a pressure is reached below which no flame will propagate. However, this behavior appears to be due to wall quenching by the tube in which the experiments were performed. As shown in Fig. 4.26, the limits are actually as wide at low pressure as at 1 atm, provided the tube is sufficiently wide and an ignition source can be found to

Natural gas (% by volume)

60 50 40

Flammable area

30 20 10 0

0

800

1600

2000

2400

Pressure (Ib/in2 gauge) FIGURE 4.25 Effect of pressure increase above atmospheric pressure on flammability limits of natural gas–air mixtures (from Lewis and von Elbe [5]).

Pressure (mm Hg)

800

600

400

Flammable area

200

0

0

2

4 6 8 10 12 14 Natural gas (% by volume)

16

FIGURE 4.26 Effect of reduction of pressure below atmospheric pressure on flammability limits of natural gas–air mixtures (from Lewis and von Elbe [5]).

198

Combustion

ignite the mixtures. Consequently, the limits obtained at reduced pressures are not generally true limits of flammability, since they are influenced by the tube diameter. Therefore, these limits are not physicochemical constants of a given fuel. Rather, they are limits of flame propagation in a tube of specified diameter. In examining the effect of high pressures on flammability limits, it has been assumed that the limit is determined by a critical value of the rate of heat loss to the rate of heat development. Consider, for example, a flame anchored on a Bunsen tube. The loss to the anchoring position is small, and thus the radiation loss must be assumed to be the major heat loss condition. This radiative loss is in the infrared, due primarily to the band radiation systems of CO2, H2O, and CO. The amount of product composition changes owing to dissociation at the flammability limits is indeed small, so there is essentially no increase in temperature with pressure. Even so, with temperatures near the limits and wavelengths of the gaseous radiation, the radiation bands lie near or at the maximum of the energy density radiation distribution given by Planck’s law. If λ is the wavelength, then λmax T equals a constant, by Wien’s law. Thus the radiant loss varies as T 5. But for most hydrocarbon systems the activation energy of the reaction media and temperature are such that the variation of exp(E/RT) as a function of temperature is very much like a T 5 variation [30]. Thus, any effect of pressure on temperature shows a balance of these loss and gain terms, except that the actual radiation contains an emissivity term. Due to band system broadening and emitting gas concentration, this emissivity is approximately proportional to the total pressure for gaseous systems. Then, as the pressure increases, the emissivity and heat loss increase monotonically. On the fuel-rich side the reaction rate is second-order and the energy release increases with P2 as compared to the heat loss that increases with P. Thus the richness of the system can be increased as the pressure increases before extinction occurs [30]. For the methane flammability results reported in Fig. 4.25, the rich limit clearly broadens extensively and then begins to level off as the pressure is increased over a range of about 150 atm. The leveling-off happens when soot formation occurs. The soot increases the radiative loss. The lean limit appears not to change with pressure, but indeed broadens by about 25% over the same pressure range. Note that over a span of 28 atm, the rich limit broadens about 300% and the lean limit only about 1%. There is no definitive explanation of this difference; but, considering the size, it could possibly be related to the temperature because of its exponential effect on the energy release rate and the emissive power of the product gases. The rule of thumb quoted earlier that the rich limit is about 3 times the stoichiometric value and the lean limit half the stoichiometric value can be rationalized from the temperature effect. Burning near the rich limit generates mostly diatomic products—CO and H2—and some H2O. Burning near the lean limit produces CO2 and H2O exclusively. Thus for the same percentage composition change, regardless of the energy effect, the fuel-rich side temperature will be higher than the lean side temperature. As was emphasized in Chapter 1, for hydrocarbons the maximum

Flame Phenomena in Premixed Combustible Gases

199

flame temperature occurs on the fuel-rich side of stoichiometric owing to the presence of diatomics, particularly H2. Considering percentage changes due to temperature, the fuel side flammability limit can broaden more extensively as one increases the pressure to account for the reaction rate compensation necessary to create the new limit. Furthermore the radiative power of the fuelrich side products is substantially less than that of the lean side because the rich side contains only one diatomic radiator and a little water, whereas the lean side contains exclusively triatomic radiators. The fact that flammability limits have been found [29] to be different for upward and downward propagation of a flame in a cylindrical tube if the tube is large enough could be an indication that heat losses [30, 31] are not the dominant extinction mechanism specifying the limit. Directly following a discourse by Ronney [32], it is well first to emphasize that buoyancy effects are an important factor in the flammability limits measured in large cylindrical tubes. Extinction of upward-propagating flames for a given fuel–oxidizer mixture ratio is thought to be due to flame stretch at the tip of the rising hemispherical flame [33, 34]. For downward propagation, extinction is thought to be caused by a cooling, sinking layer of burned gases near the tube wall that overtakes the weakly propagating flame front whose dilution leads to extinction [35, 36]. For small tubes, heat loss to walls can be the primary cause for extinction; indeed, such wall effects can quench the flames regardless of mixture ratio. Thus, as a generalization, flammability limits in tubes are probably caused by the combined influences of heat losses to the tube wall, buoyancyinduced curvature and strain, and even Lewis number effects. Because of the difference in these mechanisms, it has been found that the downward propagation limits can sometimes be wider than the upward limits, depending upon the degree of buoyancy and Lewis number. It is interesting that experiments under microgravity conditions [37, 38] reveal that the flammability limits are different from those measured for either upward or downward propagation in tubes at normal gravity. Upon comparing theoretical predictions [30] to such experimental measurements as the propagation rate at the limit and the rate of thermal decay in the burned gases, Ronney [39] suggested that radiant heat loss is probably the dominant process leading to flame extinction at microgravity. Ronney [39] concludes that, while surprising, the completely different processes dominating flammability limits at normal gravity and microgravity are readily understandable in light of the time scales of the processes involved. He showed that the characteristic loss rate time scale for upward-propagating flames in tubes (τu), downward-propagating flames (τd), radiative losses (τr), and conductive heat losses to the wall (τc) scale as (d/g)1/2, α/g2, ρcpTf/E, and d2/α, respectively. The symbols not previously defined are d, the tube diameter; g, the gravitational acceleration; α, the thermal diffusivity; and E, the radiative heat loss per unit volume. Comparison of these time scales indicates that for any practical gas mixture, pressure, and tube diameter, it is difficult

200

Combustion

to obtain τr τu or τr τd at normal gravity; thus, radiative losses are not as important as buoyancy-induced effects under this condition. At microgravity, τu and τd are very large, but still τr must be less than τc, so radiant effects are dominant. In this situation, large tube diameters are required.

2. Quenching Distance Wall quenching affects not only flammability limits, but also ignition phenomena (see Chapter 7). The quenching diameter, dT, which is the parameter given the greatest consideration, is generally determined experimentally in the following manner. A premixed flame is established on a burner port and the gas flow is suddenly stopped. If the flame propagates down the tube into the gas supply source, a smaller tube is substituted. The tube diameter is progressively decreased until the flame cannot propagate back to the source. Thus the quenching distance, or diameter dT, is the diameter of the tube that just prevents flashback. A flame is quenched in a tube when the two mechanisms that permit flame propagation—diffusion of species and of heat—are affected. Tube walls extract heat: the smaller the tube, the greater is the surface area to volume ratio within the tube and hence the greater is the volumetric heat loss. Similarly, the smaller the tube, the greater the number of collisions of the active radical species that are destroyed. Since the condition and the material composition of the tube wall affect the rate of destruction of the active species [5], a specific analytical determination of the quenching distance is not feasible. Intuition would suggest that an inverse correlation would be obtained between flame speed and quenching diameter. Since flame speed SL varies with equivalence ratio φ, so should dT vary with φ; however, the curve of dT would be inverted compared to that of SL, as shown in Fig. 4.27. One would also expect, and it is found experimentally, that increasing the temperature would decrease the quenching distance. This trend arises because

Increasing temperature curves

dT

1.0 φ FIGURE 4.27 Variation of quenching diameter dT as a function of equivalence ratio φ and trend with initial temperature.

201

Flame Phenomena in Premixed Combustible Gases

the heat losses are reduced with respect to heat release and species are not as readily deactivated. However, sufficient data are not available to develop any specific correlation. It has been concretely established and derived theoretically [30] that quenching distance increases as pressure decreases; in fact, the correlation is almost exactly dT ∼ 1/P for many compounds. For various fuels, P sometimes has an exponent somewhat less than 1. An exponent close to 1 in the dT  1/P relationship can be explained as follows. The mean free path of gases increases as pressure decreases; thus there are more collisions with the walls and more species are deactivated. Pressure results are generally represented in the form given in Fig. 4.28, which also shows that when measuring flammability limits as a function of subatmospheric pressures, one must choose a tube diameter that is greater than the dT given for the pressure. The horizontal dot-dash line in Fig. 4.28 specifies the various flammability limits that would be obtained at a given subatmospheric pressure in tubes of different diameters.

3. Flame Stabilization (Low Velocity) In the introduction to this chapter a combustion wave was considered to be propagating in a tube. When the cold premixed gases flow in a direction opposite to the wave propagation and travel at a velocity equal to the propagation velocity (i.e., the laminar flame speed), the wave (flame) becomes stationary with respect to the containing tube. Such a flame would possess only neutral stability, and its actual position would drift [1]. If the velocity of the unburned mixture is increased, the flame will leave the tube and, in most cases, fix itself

Increasing dT

P

1.0 φ FIGURE 4.28 Effect of pressure on quenching diameter.

202

Combustion

x

FIGURE 4.29

Gas mixture streamlines through a Bunsen cone flame.

in some form at the tube exit. If the tube is in a vertical position, then a simple burner configuration, as shown in Fig. 4.29, is obtained. In essence, such burners stabilize the flame. As described earlier, these burners are so configured that the fuel and air become a homogeneous mixture before they exit the tube. The length of the tube and physical characteristics of the system are such that the gas flow is laminar in nature. In the context to be discussed here, a most important aspect of the burner is that it acts as a heat and radical sink, which stabilizes the flame at its exit under many conditions. In fact, it is the burner rim and the area close to the tube that provide the stabilization position. When the flow velocity is increased to a value greater than the flame speed, the flame becomes conical in shape. The greater the flow velocity, the smaller is the cone angle of the flame. This angle decreases so that the velocity component of the flow normal to the flame is equal to the flame speed. However, near the burner rim the flow velocity is lower than that in the center of the tube; at some point in this area the flame speed and flow velocity equalize and the flame is anchored by this point. The flame is quite close to the burner rim and its actual speed is controlled by heat and radical loss to the wall. As the flow velocity is increased, the flame edge moves further from the burner, losses to the rim decrease and the flame speed increases so that another stabilization point is reached. When the flow is such that the flame edge moves far from the rim, outside air is entrained, a lean mixture is diluted, the flame speed drops, and the flame reaches its blowoff limit. If, however, the flow velocity is gradually reduced, this configuration reaches a condition in which the flame speed is greater than the flow velocity at some point across the burner. The flame will then propagate down into the burner, so that the flashback limit is reached. Slightly before the flashback limit is reached, tilted flames may occur. This situation occurs because the back pressure of the flame causes a disturbance in the flow so that the flame can enter the burner only in the region where the flow velocity is reduced.

203

Flame Phenomena in Premixed Combustible Gases

Limit of luminous zone Flame front

Streamlines FIGURE 4.30 Formation of a tilted flame (after Bradley [1]).

Because of the constraint provided by the burner tube, the flow there is less prone to distortion; so further propagation is prevented and a tilted flame such as that shown in Fig. 4.30 is established [1]. Thus it is seen that the laminar flame is stabilized on burners only within certain flow velocity limits. The following subsections treat the physical picture just given in more detail.

a. Flashback and Blowoff Assume Poiseuille flow in the burner tube. The gas velocity is zero at the stream boundary (wall) and increases to a maximum in the center of the stream. The linear dimensions of the wall region of interest are usually very small; in slow burning mixtures such as methane and air, they are of the order of 1 mm. Since the burner tube diameter is usually large in comparison, as shown in Fig. 4.31, the gas velocity near the wall can be represented by an approximately linear vector profile. Figure 4.31 represents the conditions in the area where the flame is anchored by the burner rim. Further assume that the flow lines of the fuel jet are parallel to the tube axis, that a combustion wave is formed in the stream, and that the fringe of the wave approaches the burner rim closely. Along the flame wave profile, the burning velocity attains its maximum value SL0 . Toward the fringe, the burning velocity decreases as heat and chain carriers are lost to the rim. If the wave fringe is very close to the rim (position

204

Combustion

S 0u S 0u S 0u Velocity profile of gas

3 Open atmosphere

2 1

Solid edge FIGURE 4.31

Stabilization positions of a Bunsen burner flame (after Lewis and von Elbe [5]).

u1

SL

u2 u3 SL u x dp FIGURE 4.32 General burning velocity and gas velocity profiles inside a Bunsen burner tube (from Lewis and von Elbe [5]).

1 in Fig. 4.31), the burning velocity in any flow streamline is smaller than the gas velocity and the wave is driven farther away by gas flow. As the distance from the rim increases, the loss of heat and chain carriers decreases and the burning velocity becomes larger. Eventually, a position is reached (position 2 in Fig. 4.31) in which the burning velocity is equal to the gas velocity at some point of the wave profile. The wave is now in equilibrium with respect to the solid rim. If the wave is displaced to a larger distance (position 3 in Fig. 4.31), the burning velocity at the indicated point becomes larger than the gas velocity and the wave moves back to the equilibrium position. Consider Fig. 4.32, a graph of flame velocity SL as a function of distance, for a wave inside a tube. In this case, the flame has entered the tube. The distance from the burner wall is called the penetration distance dp (half the quenching diameter dT). If u1 is the mean velocity of the gas flow in the tube and the line labeled u1 is the graph of the velocity profile near the tube wall, the local flame velocity is not greater than the local gas velocity at any point; therefore, any flame that finds itself inside the tube will then blow out of the tube. At a lower velocity u2 , which is just tangent to the SL curve, a stable point is reached. Then u2 is the minimum mean velocity before flashback occurs. The line for the mean velocity u3 indicates a region where the flame speed is greater than the velocity in the tube represented by u3 ; in this case,

205

Flame Phenomena in Premixed Combustible Gases

u4

u3

SL

4

u2

3

u1

2

SL 1

2 4 3

u

Distance from boundary of stream

1

Atmosphere Solid edge

FIGURE 4.33 Burning velocity and gas velocity profiles above a Bunsen burner tube rim (from Lewis and von Elbe [5]).

the flame does flash back. The gradient for flashback, gF, is SL/dp. Analytical developments [30] show that dp ≈ (λ /c p ρ )(1/SL ) ≈ (α /SL ) Similar reasoning can apply to blowoff, but the arguments are somewhat different and less exact because nothing similar to a boundary layer exists. However, a free boundary does exist. When the gas flow in the tube is increased, the equilibrium position shifts away from the rim. With increasing distance from the rim, a lean gas mixture becomes progressively diluted by interdiffusion with the surrounding atmosphere, and the burning velocity in the outermost streamlines decreases correspondingly. This effect is indicated by the increasing retraction of the wave fringe for flame positions 1–3 in Fig. 4.33. But, as the wave moves farther from the rim, it loses less heat and fewer radicals to the rim, so it can extend closer to the hypothetical edge. However, an ultimate equilibrium position of the wave exists beyond which the effect of dilution overbalances the effect of increased distance from the burner rim everywhere on the burning velocity. If the boundary layer velocity gradient is so large that the combustion wave is driven beyond this position, the gas velocity exceeds the burning velocity along every streamline and the combustion wave blows off. These trends are represented diagrammatically in Fig. 4.33. The diagram follows the postulated trends in which SL0 is the flame velocity after the gas has been diluted because the flame front has moved slightly past u3 . Thus, there is blowoff and u3 is the blowoff velocity.

b. Analysis and Results The topic of concern here is the stability of laminar flames fixed to burner tubes. The flow profile of the premixed gases flowing up the tube in such a system must be parabolic; that is, Poiseuille flow exists. The gas velocity along any streamline is given by u  n(R 2  r 2)

206

Combustion

where R is the tube radius. Since the volumetric flow rate, Q (cm3/s) is given by R

Q  ∫ 2πru dr 0

then n must equal n  2Q/π R 4 The gradient for blowoff or flashback is defined as gF,B ≡ lim (du/dr ) r→R then gF,B 

u u 4Q  4 av  8 av 3 πR R d

where d is the diameter of the tube. Most experimental data on flashback are plotted as a function of the average flashback velocity, uav,F, as shown in Fig. 4.34. It is possible to estimate penetration distance (quenching thickness) from the burner wall in graphs such as Fig. 4.34 by observing the cut-off radius for each mixture. The development for the gradients of flashback and blowoff suggests a more appropriate plot of gB,F versus φ, as shown in Figs. 4.35 and 4.36. Examination of these figures reveals that the blowoff curve is much steeper than that for flashback. For rich mixtures the blowoff curves continue to rise instead of decreasing after the stoichiometric value is reached. The reason for this trend is that experiments are performed in air, and the diffusion of air into

Increasing tube diameter uav,F

1.0 φ FIGURE 4.34

Critical flow for flashback as a function of equivalence ratio φ·

207

Flame Phenomena in Premixed Combustible Gases

the mixture as the flame lifts off the burner wall increases the local flame speed of the initially fuel-rich mixture. Experiments in which the surrounding atmosphere was not air, but nitrogen, verify this explanation and show that the gB would peak at stoichiometric. The characteristics of the lifted flame are worthy of note as well. Indeed, there are limits similar to those of the seated flame [1]. When a flame is in the lifted position, a dropback takes place when the gas velocity is reduced, and the flame takes up its normal position on the burner rim. When the gas velocity is increased instead, the flame will blow out. The instability requirements of both the seated and lifted flames are shown in Fig. 4.37.

4. Stability Limits and Design The practicality of understanding stability limits is uniquely obvious when one considers the design of Bunsen tubes and cooking stoves using gaseous fuels. In the design of a Bunsen burner, it is desirable to have the maximum range of volumetric flow without encountering stability problems. What, then, is the

gF/s

500

1.0 φ FIGURE 4.35 Typical curve of the gradient of flashback as a function of equivalence ratio φ. The value of φ  1 is for natural gas.

in air gB/s

1000 in nitrogen

1.0 φ FIGURE 4.36 Typical curves of the gradient of blowoff as a function of equivalence ratio φ. The value at φ  1 is for natural gas.

208

Combustion

Blowout

Flow velocity

Lifted flames Extinction

Lift

Seated or lifted flames

Blowoff

Dropback Seated flames Flashback

Fuel–oxygen ratio FIGURE 4.37

Seated and lifted flame regimes for Bunsen-type burners.

optimum size tube for maximum flexibility? First, the diameter of the tube must be at least twice the penetration distance, that is, greater than the quenching distance. Second, the average velocity must be at least twice SL; otherwise, a precise Bunsen cone would not form. Experimental evidence shows further that if the average velocity is 5 times SL, the fuel penetrates the Bunsen cone tip. If the Reynolds number of the combustible gases in the tube exceeds 2000, the flow can become turbulent, destroying the laminar characteristics of the flame. Of course, there are the limitations of the gradients of flashback and blowoff. If one graphs uav versus d for these various limitations, one obtains a plot similar to that shown in Fig. 4.38. In this figure the dotted area represents the region that has the greatest flow variability without some stabilization problem. Note that this region d maximizes at about 1 cm; consequently, the tube diameter of Bunsen burners is always about 1 cm. The burners on cooking stoves are very much like Bunsen tubes. The fuel induces air and the two gases premix prior to reaching the burner ring with its flame holes. It is possible to idealize this situation as an ejector. For an ejector, the total gas mixture flow rate can be related to the rate of fuel admitted to the system through continuity of momentum mm um  mf uf um (ρm Am um )  (ρf Af uf )uf

209

Flame Phenomena in Premixed Combustible Gases

uavd   2000

uav 

gB d 8

5SL

uav 

gF d 8

uav 2SL

2dp

1 cm Tube diameter (d)

FIGURE 4.38 Stability and operation limits of a Bunsen burner.

Stagnant air

Fuel

Af

Am

FIGURE 4.39 Fuel jet ejector system for premixed fuel–air burners.

where the subscript m represents the conditions for the mixture (Am is the total area) and the subscript f represents conditions for the fuel. The ejector is depicted in Fig. 4.39. The momentum expression can be written as ρm um2  aρf uf2 where a is the area ratio. If one examines the gF and gB on the graph shown in Fig. 4.40, one can make some interesting observations. The burner port diameter is fixed such that a rich mixture ratio is obtained at a value represented by the dashed line on Fig. 4.40. When the mixture ratio is set at this value, the flame can never flash back into the stove and burn without the operator’s noticing the situation. If the fuel is changed, difficulties may arise. Such difficulties arose many decades ago when the gas industry switched from manufacturer’s gas to natural gas, and could arise again if the industry is ever compelled to

210

Combustion

gB

gF gF

gB

1.0 φ FIGURE 4.40

Flame stability diagram for an operating fuel gas–air mixture burner system.

switch to synthetic gas or to use synthetic or petroleum gas as an additive to natural gas. The volumetric fuel–air ratio in the ejector is given by (F/A)  (uf Af )/(um Am ) It is assumed here that the fuel–air (F/A) mixture is essentially air; that is, the density of the mixture does not change as the amount of fuel changes. From the momentum equation, this fuel–air mixture ratio becomes (F/A)  (ρm /ρf )1/ 2 a1/ 2 The stoichiometric molar (volumetric) fuel–air ratio is strictly proportional to the molecular weight of the fuel for two common hydrocarbon fuels; that is, (F/A)stoich ∼ 1/MWf ∼ 1/ρf The equivalence ratio then is φ

a1/ 2 (ρm /ρf )1/ 2 (F/A) ∼ (F/A)stoich (1/ρf )

Examining Fig. 4.40, one observes that in converting from a heavier fuel to a lighter fuel, the equivalence ratio drops, as indicated by the new dot-dash operating line. Someone adjusting the same burner with the new lighter fuel would have a very consistent flashback–blowoff problem. Thus, when the gas industry switched to natural gas, it was required that every fuel port in every burner on every stove be drilled open, thereby increasing a to compensate for the decreased ρf. Synthetic gases of the future will certainly be heavier than methane (natural gas). They will probably be mostly methane with some heavier

Flame Phenomena in Premixed Combustible Gases

211

components, particularly ethane. Consequently, today’s burners will not present a stability problem; however, they will operate more fuel-rich and thus be more wasteful of energy. It would be logical to make the fuel ports smaller by inserting caps so that the operating line would be moved next to the rich flashback cut-off line.

E. FLAME PROPAGATION THROUGH STRATIFIED COMBUSTIBLE MIXTURES Liquid fuel spills occur under various circumstances and, in the presence of an ignition source, a flame can be established and propagate across the surface. In a stagnant atmosphere how the flame propagates through the combustible mixture of the fuel vapor and air is strongly dependent on the liquid fuel’s temperature. The relative tendency of a liquid fuel to ignite and propagate is measured by various empirical techniques. Under the stagnant atmosphere situation a vapor pressure develops over the liquid surface and a stratified fuel vapor–air mixture develops. At a fixed distance above the liquid an ignition source is established and the temperature of the fuel is raised until a flame flashes. This procedure determines the so-called flash point temperature [40]. After the ignition there generally can be no flame propagation. The point at which a further increase of the liquid temperature causes flame propagation over the complete fuel surface is called the fire point. The differences in temperature between the flash and fire points are generally very slight. The stratified gaseous layer established over the liquid fuel surface varies from a fuel-rich mixture to within the lean flammability limits of the vaporized fuel and air mixture. At some point above the liquid surface, if the fuel temperature is high enough, a condition corresponds to a stoichiometric equivalence ratio. For most volatile fuels this stoichiometric condition develops. Experimental evidence indicates that the propagation rate of the curved flame front that develops is many times faster than the laminar flame speed discussed earlier. There are many less volatile fuels, however, that only progress at very low rates. It is interesting to note that stratified combustible gas mixtures can exist in tunnel-like conditions. The condition in a coal mine tunnel is an excellent example. The marsh gas (methane) is lighter than air and accumulates at the ceiling. Thus a stratified air–methane mixture exists. Experiments have shown that under the conditions described the flame propagation rate is very much faster than the stoichiometric laminar flame speed. In laboratory experiments simulating the mine-like conditions the actual rates were found to be affected by the laboratory simulated tunnel length and depth. In effect, the expansion of the reaction products of these type laboratory experiments drives the flame front developed. The overall effect is similar in context to the soap bubble type flame experiments discussed in Section C5c. In the soap bubble flame experiment measurements, the ambient condition is about 300 K and the stoichiometric flame temperature of the flame products for most hydrocarbon fuels

212

Combustion

is somewhat above 2200 K, so that the observed flame propagation rate in the soap bubble is 7–8 times the laminar flame speed. Thus in the soap bubble experiment the burned gases drive the flame front and, of course, a small differential pressure exists across this front. Under the conditions described for coal mine tunnel configurations the burned gas expansion effect develops differently. To show this effect Feng et al. [41, 42] considered analytically the propagation of a fuel layer at the roof of a channel over various lengths and depths of the configuration in which the bottom layer was simply air. For the idealized infinite depth of the air layer the results revealed that the ratio of the propagating flame speed to that of the laminar flame speed was equal to the square root of the density ratio (ρu/ρb); that is, the flame propagation for the layered configuration is about 2.6–2.8 times the laminar flame speed. Indeed the observed experimental trends [41, 42] fit the analytical derivations. The same trends appear to hold for the case of a completely premixed combustible condition of the roof of a channel separated from the air layer below [41]. The physical perception derived from these analytical results was that the increased flame propagation speed over the normal flame speed was due to a fluid dynamical interaction resulting from the combustion of premixed gases; that is, after the combustible gas mixture moves through the flame front, the expansion of the product gases causes a displacement of the unburned gases ahead of the flame. This displacement results in redistributing the combustible gaseous layer over a much larger area in the induced curved, parabolic type, flame front created. Thus the expansion of the combustible mixture sustains a pressure difference across the flame and the resulting larger combustible gas area exposed to the flame front increases the burning rate necessary for the elevated flame propagation rate [40, 42]. The inverse of the tunnel experiments discussed is the propagation of a flame across a layer of a liquid fuel that has a low flash point temperature. The stratified conditions discussed previously described the layered fuel vapor–air mixture ratios. Under these conditions the propagation rates were found to be 4–5 times the laminar flame speed. This somewhat increased rate compared to the other analytical results is apparently due to diffusion of air to the flame front behind the parabolic leading edge of the propagating flame [41]. Experiments [43] with very high flash point fuels (JP, kerosene, Diesel, etc.) revealed that the flame propagation occurred in an unusual manner and a much slower rate. In this situation, at ambient conditions, any possible amount of fuel vapor above the liquid surface creates a gaseous mixture well outside the fuel’s flammability limits. What was discovered [44, 45] was that for these fuels the flame will propagate due to the fact that the liquid surface under the ignition source is raised to a local temperature that is higher than the cool ambient temperature ahead of the initiated flame. Experimental observations revealed [45] that this surface temperature variation from behind the flame front to the cool region ahead caused a variation in the surface tension

Flame Phenomena in Premixed Combustible Gases

213

of the liquid fuel. Since the surface tension of a liquid varies inversely with the temperature, a gradient in surface tension is established and creates a surface velocity from the warmer temperature to the cooler temperature. Thus volatile liquid is pulled ahead of the flame front to provide the combustible vapor–air mixture for flame propagation. Since the liquid is a viscous fluid, currents are established throughout the liquid fluid layer by the surface movement caused by the surface tension variation. In the simplest context of thin liquid fuel films the problem of estimating the velocities in the liquid is very much like the Couette flow problem, except that movement of the viscous liquid fuel is not established by a moving plate, but by the surface tension variation along the free surface. Under such conditions at the surface the following equality exists: τ  μ(∂u/∂y)s  σ x  (d σ/dT )(dT/dx ) where τ is the shear stress in the liquid, μ is the liquid fuel viscosity, u the velocity parallel to the surface, y the direction normal to the surface, s the surface point, σ the surface tension, T the temperature, and x the direction along the surface. The following proportionality is readily developed from the above equation: us ∼ σ x h/μ where us the surface velocity and h is the depth of the fuel layer. In some experiments the viscosity of a fuel was varied by addition of a thickening agent that did not affect the fuel volatility [40]. For a fixed fuel depth it was found that the flame propagation rate varied inversely with the induced viscosity as noted by the above proportionality. Because the surface-tension-induced velocity separates any liquid fuel in front of the initiated induced flame, very thin fuel layers do not propagate flames [40]. For very deep fuel layers the Couette flow condition does not hold explicitly and an inverted boundary layer type flow exists in the liquid as the flame propagates. Many nuances with respect to the observed flame propagation for physical conditions varied experimentally can be found in the reference detailed by Ref. [40]. Propagation across solid fuel surfaces is a much more complex problem because the orientation of the solid surface can be varied. For example, a sheet of plastic or wood held in a vertical position and ignited at either the top or bottom edge shows vastly different propagation rate because of gravity effects. Even material held at an angle has a different burning rate than the two possible vertical conditions. A review of this solid surface problem can be found in Refs. [45a, 45b, 45c].

F. TURBULENT REACTING FLOWS AND TURBULENT FLAMES Most practical combustion devices create flow conditions so that the fluid state of the fuel and oxidizer, or fuel–oxidizer mixture, is turbulent. Nearly all

214

Combustion

mobile and stationary power plants operate in this manner because turbulence increases the mass consumption rate of the reactants, or reactant mixture, to values much greater than those that can be obtained with laminar flames. A greater mass consumption rate increases the chemical energy release rate and hence the power available from a given combustor or internal engine of a given size. Indeed, few combustion engines could function without the increase in mass consumption during combustion that is brought about by turbulence. Another example of the importance of turbulence arises with respect to spark timing in automotive engines. As the RPM of the engine increases, the level of turbulence increases, whereupon the mass consumption rate (or turbulent flame speed) of the fuel–air mixture increases. This explains why spark timing does not have to be altered as the RPM of the engine changes with a given driving cycle. As has been shown, the mass consumption rate per unit area in premixed laminar flames is simply ρSL, where ρ is the unburned gas mixture density. Correspondingly, for power plants operating under turbulent conditions, a similar consumption rate is specified as ρST, where ST is the turbulent burning velocity. Whether a well-defined turbulent burning velocity characteristic of a given combustible mixture exists as SL does under laminar conditions will be discussed later in this section. What is known is that the mass consumption rate of a given mixture varies with the state of turbulence created in the combustor. Explicit expressions for a turbulent burning velocity ST will be developed, and these expressions will show that various turbulent fields increase ST to values much larger than SL. However, increasing turbulence levels beyond a certain value increases ST very little, if at all, and may lead to quenching of the flame [46]. To examine the effect of turbulence on flames, and hence the mass consumption rate of the fuel mixture, it is best to first recall the tacit assumption that in laminar flames the flow conditions alter neither the chemical mechanism nor the associated chemical energy release rate. Now one must acknowledge that, in many flow configurations, there can be an interaction between the character of the flow and the reaction chemistry. When a flow becomes turbulent, there are fluctuating components of velocity, temperature, density, pressure, and concentration. The degree to which such components affect the chemical reactions, heat release rate, and flame structure in a combustion system depends upon the relative characteristic times associated with each of these individual parameters. In a general sense, if the characteristic time (τc) of the chemical reaction is much shorter than a characteristic time (τm) associated with the fluid-mechanical fluctuations, the chemistry is essentially unaffected by the flow field. But if the contra condition (τc τm) is true, the fluid mechanics could influence the chemical reaction rate, energy release rates, and flame structure. The interaction of turbulence and chemistry, which constitutes the field of turbulent reacting flows, is of importance whether flame structures exist or not.

Flame Phenomena in Premixed Combustible Gases

215

The concept of turbulent reacting flows encompasses many different meanings and depends on the interaction range, which is governed by the overall character of the flow environment. Associated with various flows are different characteristic times, or, as more commonly used, different characteristic lengths. There are many different aspects to the field of turbulent reacting flows. Consider, for example, the effect of turbulence on the rate of an exothermic reaction typical of those occurring in a turbulent flow reactor. Here, the fluctuating temperatures and concentrations could affect the chemical reaction and heat release rates. Then, there is the situation in which combustion products are rapidly mixed with reactants in a time much shorter than the chemical reaction time. (This latter example is the so-called stirred reactor, which will be discussed in more detail in the next section.) In both of these examples, no flame structure is considered to exist. Turbulence-chemistry interactions related to premixed flames comprise another major stability category. A turbulent flow field dominated by largescale, low-intensity turbulence will affect a premixed laminar flame so that it appears as a wrinkled laminar flame. The flame would be contiguous throughout the front. As the intensity of turbulence increases, the contiguous flame front is partially destroyed and laminar flamelets exist within turbulent eddies. Finally, at very high-intensity turbulence, all laminar flame structure disappears and one has a distributed reaction zone. Time-averaged photographs of these three flames show a very bushy flame front that looks very thick in comparison to the smooth thin zone that characterizes a laminar flame. However, when a very fast response thermocouple is inserted into these three flames, the fluctuating temperatures in the first two cases show a bimodal probability density function with well-defined peaks at the temperatures of the unburned and completely burned gas mixtures. But a bimodal function is not found for the distributed reaction case. Under premixed fuel–oxidizer conditions the turbulent flow field causes a mixing between the different fluid elements, so the characteristic time was given the symbol τm. In general with increasing turbulent intensity, this time approaches the chemical time, and the associated length approaches the flame or reaction zone thickness. Essentially the same is true with respect to nonpremixed flames. The fuel and oxidizer (reactants) in non-premixed flames are not in the same flow stream; and, since different streams can have different velocities, a gross shear effect can take place and coherent structures (eddies) can develop throughout this mixing layer. These eddies enhance the mixing of fuel and oxidizer. The same type of shear can occur under turbulent premixed conditions when large velocity gradients exist. The complexity of the turbulent reacting flow problem is such that it is best to deal first with the effect of a turbulent field on an exothermic reaction in a plug flow reactor. Then the different turbulent reacting flow regimes will be described more precisely in terms of appropriate characteristic lengths, which will be developed from a general discussion of turbulence. Finally, the turbulent premixed flame will be examined in detail.

216

Combustion

1. The Rate of Reaction in a Turbulent Field As an excellent, simple example of how fluctuating parameters can affect a reacting system, one can examine how the mean rate of a reaction would differ from the rate evaluated at the mean properties when there are no correlations among these properties. In flow reactors, time-averaged concentrations and temperatures are usually measured, and then rates are determined from these quantities. Only by optical techniques or very fast response thermocouples could the proper instantaneous rate values be measured, and these would fluctuate with time. The fractional rate of change of a reactant can be written as ω  k ρ n1Yin  AeE/RT (P/R )n1 T 1nYin where the Yi ’s are the mass fractions of the reactants. The instantaneous change in rate is given by d ω  A(P/R)n1 ⎡⎣ (E/RT 2 )eE/RT T 1nYin dT  (1  n) T n eE/RT Yin dT  neE/RT T 1nYin1 dY ⎤⎦  (dYi /Yi ) d ω  (E/RT )ω (dT/T )  (1  n)ω (dT/T )  ωn or d ω /ω  [E/RT  (1  n)] (dT/T ) + n(dYi /Yi ) For most hydrocarbon flame or reacting systems the overall order of reaction is about 2, E/R is approximately 20,000 K, and the flame temperature is about 2000 K. Thus, (E/RT )  (1  n) ≅ 9 and it would appear that the temperature variation is the dominant factor. Since the temperature effect comes into this problem through the specific reaction rate constant, the problem simplifies to whether the mean rate constant can be represented by the rate constant evaluated at the mean temperature. In this hypothetical simplified problem one assumes further that the temperature T fluctuates with time around some mean represented by the form T (t )/T  1  an f (t ) where an is the amplitude of the fluctuation and f(t) is some time-varying function in which 1  f (t )  1

Flame Phenomena in Premixed Combustible Gases

217

and T 

1 τ

τ

∫ T (t )dt 0

over some time interval τ. T(t) can be considered to be composed of T  T (t ) , where T is the fluctuating component around the mean. Ignoring the temperature dependence in the pre-exponential, one writes the instantaneous–rate constant as k (T )  A exp(E/RT ) and the rate constant evaluated at the mean temperature as k (T )  A exp( E/RT ) Dividing the two expressions, one obtains {k (T )/k (T )}  exp {( E/RT )[1  (T/T )]} Obviously, then, for small fluctuations 1  (T/T ) = [ an f (t )]/[1  an f (t )] ≈ an f (t ) The expression for the mean rate is written as τ

1 k (T )  τ k (T )

∫

1 τ

∫

=

0 τ

0

τ ⎞ ⎛ E 1 k (T ) dt  ∫ exp ⎜⎜ an f (t ) ⎟⎟⎟ dt ⎜⎝ RT ⎠ τ 0 k (T ) 2 ⎡ ⎤ ⎢1  E a f (t )  1 ⎛⎜ E a f (t ) ⎞⎟⎟  ⎥ dt ⎜⎜ ⎢ ⎥ n n ⎟⎠ RT 2 ⎝ RT ⎢⎣ ⎥⎦

But recall τ

∫

f (t )dt  0

and

0  f 2 (t )  1

0

Examining the third term, it is apparent that 1 τ

τ

∫ 0

an2 f 2 (t )dt  an2

218

Combustion

since the integral of the function can never be greater than 1. Thus, ⎞2 k (T ) 1⎛ E  1  ⎜⎜ an ⎟⎟ k (T ) 2 ⎜⎝ RT ⎟⎠

Δ=

or

⎞2 k (T )  k (T ) 1⎛ E  ⎜⎜ an ⎟⎟ k (T ) 2 ⎜⎝ RT ⎟⎠

If the amplitude of the temperature fluctuations is of the order of 10% of the mean temperature, one can take an  0.1; and if the fluctuations are considered sinusoidal, then 1 τ

τ

1

∫ sin2 t dt  2 0

Thus for the example being discussed, Δ

2 ⎞⎟2 1 ⎛⎜ E 1 ⎛⎜ 40, 000  0.1 ⎞⎟ a  ⎟ , ⎜ ⎜ n⎟ 4 ⎜⎝ RT ⎟⎠ 4 ⎜⎝ 2  2000 ⎟⎠

Δ≅

1 4

or a 25% difference in the two rate constants. This result could be improved by assuming a more appropriate distribution function of T instead of a simple sinusoidal fluctuation; however, this example—even with its assumptions—usefully illustrates the problem. Normally, probability distribution functions are chosen. If the concentrations and temperatures are correlated, the rate expression becomes very complicated. Bilger [47] has presented a form of a two-component mean-reaction rate when it is expanded about the mean states, as follows:

{

ω  ρ 2YiY j exp(E/RT ) 1  (ρ2 / ρ 2 )}  (YiY j )/(YiY j )  2(ρYi /ρ Yi )  2(ρY j /ρ Y j )  (E/RT ) (YiT  /YiY ) (Y j T/Y j T )

}

[(E/ 2 RT )  1] (T 2 /T 2 ) +

2. Regimes of Turbulent Reacting Flows The previous example epitomizes how the reacting media can be affected by a turbulent field. To understand the detailed effect, one must understand the elements of the field of turbulence. When considering turbulent combustion systems in this regard, a suitable starting point is the consideration of the quantities that determine the fluid characteristics of the system. The material presented subsequently has been mostly synthesized from Refs. [48] and [49]. Most flows have at least one characteristic velocity, U, and one characteristic length scale, L, of the device in which the flow takes place. In addition there is at

Flame Phenomena in Premixed Combustible Gases

219

least one representative density ρ0 and one characteristic temperature T0, usually the unburned condition when considering combustion phenomena. Thus, a characteristic kinematic viscosity ν0  μ0 /ρ0 can be defined, where μ0 is the coefficient of viscosity at the characteristic temperature T0. The Reynolds number for the system is then Re  UL/ν0. It is interesting that ν is approximately proportional to T2. Thus, a change in temperature by a factor of 3 or more, quite modest by combustion standards, means a drop in Re by an order of magnitude. Thus, energy release can damp turbulent fluctuations. The kinematic viscosity ν is inversely proportional to the pressure P, and changes in P are usually small; the effects of such changes in ν typically are much less than those of changes in T. Even though the Reynolds number gives some measure of turbulent phenomena, flow quantities characteristic of turbulence itself are of more direct relevance to modeling turbulent reacting systems. The turbulent kinetic energy q may be assigned a representative value q0 at a suitable reference point. The relative intensity of the turbulence is then characterized by either q0 /(1/2 U 2 ) or U/U, where U = (2q0 )1/2 is a representative root-mean-square velocity fluctuation. Weak turbulence corresponds to U/U 1 and intense turbulence has U/U of the order unity. Although a continuous distribution of length scales is associated with the turbulent fluctuations of velocity components and of state variables (P, ρ, T ), it is useful to focus on two widely disparate lengths that determine separate effects in turbulent flows. First, there is a length l0, which characterizes the large eddies, those of low frequencies and long wavelengths; this length is sometimes referred to as the integral scale. Experimentally, l0 can be defined as a length beyond which various fluid-mechanical quantities become essentially uncorrelated; typically, l0 is less than L but of the same order of magnitude. This length can be used in conjunction with U to define a turbulent Reynolds number Rl  Ul0 /ν 0 which has more direct bearing on the structure of turbulence in flows than does Re. Large values of Rl can be achieved by intense turbulence, large-scale turbulence, and small values of ν produced, for example, by low temperatures or high pressures. The cascade view of turbulence dynamics is restricted to large values of Rl. From the characterization of U and l0, it is apparent that Rl Re. The second length scale characterizing turbulence is that over which molecular effects are significant; it can be introduced in terms of a representative rate of dissipation of velocity fluctuations, essentially the rate of dissipation of the turbulent kinetic energy. This rate of dissipation, which is given by the symbol ε0, is ε0 

q0 (U)2 (U)3 ≈ ≈ t (l0 /U) l0

This rate estimate corresponds to the idea that the time scale over which velocity fluctuations (turbulent kinetic energy) decay by a factor of (1/e) is

220

Combustion

the order of the turning time of a large eddy. The rate ε0 increases with turbulent kinetic energy (which is due principally to the large-scale turbulence) and decreases with increasing size of the large-scale eddies. For the small scales at which molecular dissipation occurs, the relevant parameters are the kinematic viscosity, which causes the dissipation, and the rate of dissipation. The only length scale that can be constructed from these two parameters is the so-called Kolmogorov length (scale): ⎛ ν 3 ⎞⎟1/ 4 ⎡ ⎤ 1/ 4 cm 6 s3 ⎥  (cm 4 )1/ 4  1 cm lk  ⎜⎜⎜ ⎟⎟  ⎢ ⎢ (cm 3 s3 ) (1 cm1 ) ⎥ ⎝⎜ ε0 ⎟⎠ ⎣ ⎦ However, note that lk  [ν 3l0 /(U)3 ]1/ 4  [(ν 3l04 )/(U)3 l03 ]1/ 4  (l04 /Rl3 )1/ 4 Therefore lk  l0 /Rl3 / 4 This length is representative of the dimension at which dissipation occurs and defines a cut-off of the turbulence spectrum. For large Rl there is a large spread of the two extreme lengths characterizing turbulence. This spread is reduced with the increasing temperature found in combustion of the consequent increase in ν0. Considerations analogous to those for velocity apply to scalar fields as well, and lengths analogous to lk have been introduced for these fields. They differ from lk by factors involving the Prandtl and Schmidt numbers, which differ relatively little from unity for representative gas mixtures. Therefore, to a first approximation for gases, lk may be used for all fields and there is no need to introduce any new corresponding lengths. An additional length, intermediate in size between l0 and lk, which often arises in formulations of equations for average quantities in turbulent flows is the Taylor length (λ), which is representative of the dimension over which strain occurs in a particular viscous medium. The strain can be written as (U/l0). As before, the length that can be constructed between the strain and the viscous forces is λ  ⎡⎣ ν /(U/l0 ) ⎤⎦

1/ 2

λ 2  (ν l0 /U)  (ν l02 /Ul0 )  (l02 /Rl ) and then λ  l0 /Rl1/2 In a sense, the Taylor microscale is similar to an average of the other scales, l0 and lk, but heavily weighted toward lk.

221

Flame Phenomena in Premixed Combustible Gases

Recall that there are length scales associated with laminar flame structures in reacting flows. One is the characteristic thickness of a premixed flame, δL, given by ⎛ α ⎞⎟1/ 2 ⎛ cm 2 s1 ⎞⎟1/ 2 ⎟⎟  ⎜⎜ ⎟ δL ≈ ⎜⎜⎜ ⎜⎝ 1 s1 ⎟⎟⎠  1 cm ⎝ ω /ρ ⎟⎠ The derivation is, of course, consistent with the characteristic velocity in the flame speed problem. This velocity is obviously the laminar flame speed itself, so that SL 

ν cm 2 s1   1 cm s1 δL 1 cm

As discussed in an earlier section, δL is the characteristic length of the flame and includes the thermal preheat region and that associated with the zone of rapid chemical reaction. This reaction zone is the rapid heat release flame segment at the high-temperature end of the flame. The earlier discussion of flame structure from detailed chemical kinetic mechanisms revealed that the heat release zone need not be narrow compared to the preheat zone. Nevertheless, the magnitude of δL does not change, no matter what the analysis of the flame structure is. It is then possible to specify the characteristic time of the chemical reaction in this context to be ⎛ 1 ⎞⎟ δ ⎟⎟ ≈ L τ c ≈ ⎜⎜⎜ ⎟ ⎝ ω /ρ ⎠ SL It may be expected, then, that the nature of the various turbulent flows, and indeed the structures of turbulent flames, may differ considerably and their characterization would depend on the comparison of these chemical and flow scales in a manner specified by the following inequalities and designated flame type: δL lk ; Wrinkled flame

lk δL λ; Severely wrinkled flame

λ δL l0 ; Flamelets in eddies

l0 δL Distributed reaction front

The nature, or more precisely the structure, of a particular turbulent flame implied by these inequalities cannot be exactly established at this time. The reason is that values of δL, lk, λ, or l0 cannot be explicitly measured under a given flow condition or analytically estimated. Many of the early experiments with turbulent flames appear to have operated under the condition δL lk, so the early theories that developed specified this condition in expressions for ST. The flow conditions under which δL would indeed be less than lk has been explored analytically in detail and will be discussed subsequently.

222

Combustion

To expand on the understanding of the physical nature of turbulent flames, it is also beneficial to look closely at the problem from a chemical point of view, exploring how heat release and its rate affect turbulent flame structure. One begins with the characteristic time for chemical reaction designated τc, which was defined earlier. (Note that this time would be appropriate whether a flame existed or not.) Generally, in considering turbulent reacting flows, chemical lengths are constructed to be Uτc or Uτc. Then comparison of an appropriate chemical length with a fluid dynamic length provides a nondimensional parameter that has a bearing on the relative rate of reaction. Nondimensional numbers of this type are called Damkohler numbers and are conventionally given the symbol Da. An example appropriate to the considerations here is Da  (l0 /Uτ c )  (τ m /τ c )  (l0 SL /UδL ) where τm is a mixing (turbulent) time defined as (l0/U), and the last equality in the expression applies when there is a flame structure. Following the earlier development, it is also appropriate to define another turbulent time based on the Kolmogorov scale τk  (ν/ε)1/2. For large Damkohler numbers, the chemistry is fast (i.e., reaction time is short) and reaction sheets of various wrinkled types may occur. For small Da numbers, the chemistry is slow and well-stirred flames may occur. Two other nondimensional numbers relevant to the chemical reaction aspect of this problem [42] have been introduced by Frank-Kamenetskii and others. These Frank-Kamenetskii numbers (FK) are the nondimensional heat release FK1  (Qp/cpTf), where Qp is the chemical heat release of the mixture and Tf is the flame (or reaction) temperature; and the nondimensional activation energy FK2  (Ta /Tf), where the activation temperature Ta  (EA/R). Combustion, in general, and turbulent combustion, in particular, are typically characterized by large values of these numbers. When FK1 is large, chemistry is likely to have a large influence on turbulence. When FK2 is large, the rate of reaction depends strongly on the temperature. It is usually true that the larger the FK2, the thinner will be the region in which the principal chemistry occurs. Thus, irrespective of the value of the Damkohler number, reaction zones tend to be found in thin, convoluted sheets in turbulent flows, for both premixed and non-premixed systems having large FK2. For premixed flames, the thickness of the reaction region has been shown to be of the order δL/FK2. Different relative sizes of δL/FK2 and fluid-mechanical lengths, therefore, may introduce additional classes of turbulent reacting flows. The flames themselves can alter the turbulence. In simple open Bunsen flames whose tube Reynolds number indicates that the flow is in the turbulent regime, some results have shown that the temperature effects on the viscosity are such that the resulting flame structure is completely laminar. Similarly, for a completely laminar flow in which a simple wire is oscillated near the flame surface, a wrinkled flame can be obtained (Fig. 4.41). Certainly, this example is relevant to δL lk; that is, a wrinkled flame. Nevertheless, most open flames

Flame Phenomena in Premixed Combustible Gases

223

FIGURE 4.41 Flow turbulence induced by a vibrating wire. Spark shadowgraph of 5.6% propane–air flame [after Markstein, Proc. Combust. Inst. 7, 289 (1959)].

created by a turbulent fuel jet exhibit a wrinkled flame type of structure. Indeed, short-duration Schlieren photographs suggest that these flames have continuous surfaces. Measurements of flames such as that shown in Figs. 4.42a and 4.42b have been taken at different time intervals and the instantaneous flame shapes verify the continuous wrinkled flame structure. A plot of these instantaneous surface measurements results in a thick flame region (Fig. 4.43), just as the eye would visualize that a larger number of these measurements would result in a thick flame. Indeed, turbulent premixed flames are described as bushy flames. The thickness of this turbulent flame zone appears to be related to the scale of turbulence. Essentially, this case becomes that of severe wrinkling and is categorized by lk δL λ. Increased turbulence changes the character of the flame wrinkling, and flamelets begin to form. These flame elements take on the character of a fluid-mechanical vortex rather than a simple distorted wrinkled front, and this case is specified by λ δL , l0. For δL

l0, some of the flamelets fragment from the front and the flame zone becomes highly wrinkled with pockets of combustion. To this point, the flame is considered practically contiguous. When l0 δL, contiguous flames no longer exist and a distributed reaction front forms. Under these conditions, the fluid mixing processes are very rapid with respect to the chemical reaction time and the

224

(a)

Combustion

(b)

FIGURE 4.42 Short durations in Schlieren photographs of open turbulent flames [after Fox and Weinberg, Proc. Roy. Soc. Lond., A 268, 222 (1962)].

FIGURE 4.43 Superimposed contours of instantaneous flame boundaries in a turbulent flame [after Fox and Weinberg, Proc. Roy. Soc. Lond., A 268, 222 (1962)].

Flame Phenomena in Premixed Combustible Gases

225

reaction zone essentially approaches the condition of a stirred reactor. In such a reaction zone, products and reactants are continuously intermixed. For a better understanding of this type of flame occurrence and for more explicit conditions that define each of these turbulent flame types, it is necessary to introduce the flame stretch concept. This will be done shortly, at which time the regions will be more clearly defined with respect to chemical and flow rates with a graph that relates the nondimensional turbulent intensity, Reynolds numbers, Damkohler number, and characteristic lengths l. First, however, consider that in turbulent Bunsen flames the axial component of the mean velocity along the centerline remains almost constant with height above the burner; but away from the centerline, the axial mean velocity increases with height. The radial outflow component increases with distance from the centerline and reaches a peak outside the flame. Both axial and radial components of turbulent velocity fluctuations show a complex variation with position and include peaks and troughs in the flame zone. Thus, there are indications of both generation and removal of turbulence within the flame. With increasing height above the burner, the Reynolds shear stress decays from that corresponding to an initial pipe flow profile. In all flames there is a large increase in velocity as the gases enter the burned gas state. Thus, it should not be surprising that the heat release itself can play a role in inducing turbulence. Such velocity changes in a fixed combustion configuration can cause shear effects that contribute to the turbulence phenomenon. There is no better example of some of these aspects than the case in which turbulent flames are stabilized in ducted systems. The mean axial velocity field of ducted flames involves considerable acceleration resulting from gas expansion engendered by heat release. Typically, the axial velocity of the unburned gas doubles before it is entrained into the flame, and the velocity at the centerline at least doubles again. Large mean velocity gradients are therefore produced. The streamlines in the unburned gas are deflected away from the flame. The growth of axial turbulence in the flame zone of these ducted systems is attributed to the mean velocity gradient resulting from the combustion. The production of turbulence energy by shear depends on the product of the mean velocity gradient and the Reynolds stress. Such stresses provide the most plausible mechanism for the modest growth in turbulence observed. Now it is important to stress that, whereas the laminar flame speed is a unique thermochemical property of a fuel–oxidizer mixture ratio, a turbulent flame speed is a function not only of the fuel–oxidizer mixture ratio, but also of the flow characteristics and experimental configuration. Thus, one encounters great difficulty in correlating the experimental data of various investigators. In a sense, there is no flame speed in a turbulent stream. Essentially, as a flow field is made turbulent for a given experimental configuration, the mass consumption rate (and hence the rate of energy release) of the fuel–oxidizer mixture increases. Therefore, some researchers have found it convenient to define a turbulent flame speed ST as the mean mass flux per unit area (in a

226

Combustion

coordinate system fixed to the time-averaged motion of the flame) divided by the unburned gas density ρ0. The area chosen is the smoothed surface of the time-averaged flame zone. However, this zone is thick and curved; thus the choice of an area near the unburned gas edge can give quite a different result than one in which a flame position is taken in the center or the burned gas side of the bushy flame. Therefore, a great deal of uncertainty is associated with the various experimental values of ST reported. Nevertheless, definite trends have been reported. These trends can be summarized as follows: 1. ST is always greater than SL. This trend would be expected once the increased area of the turbulent flame allows greater total mass consumption. 2. ST increases with increasing intensity of turbulence ahead of the flame. Many have found the relationship to be approximately linear. (This point will be discussed later.) 3. Some experiments show ST to be insensitive to the scale of the approach flow turbulence. 4. In open flames, the variation of ST with composition is generally much the same as for SL, and ST has a well-defined maximum close to stoichiometric. Thus, many report turbulent flame speed data as the ratio of ST/SL. 5. Very large values of ST may be observed in ducted burners at high approach flow velocities. Under these conditions, ST increases in proportion to the approach flow velocities, but is insensitive to approach flow turbulence and composition. It is believed that these effects result from the dominant influence of turbulence generated within the stabilized flame by the large velocity gradients. The definition of the flame speed as the mass flux through the flame per unit area of the flame divided by the unburned gas density ρ0 is useful for turbulent nonstationary and oblique flames as well. Now with regard to stretch, consider first a plane oblique flame. Because of the increase in velocity demanded by continuity, a streamline through such an oblique flame is deflected toward the direction of the normal to the flame surface. The velocity vector may be broken up into a component normal to the flame wave and a component tangential to the wave (Fig. 4.44). Because of the energy release, the continuity of mass requires that the normal component Flame

Flow

FIGURE 4.44

Deflection of the velocity vector through an oblique flame.

Flame Phenomena in Premixed Combustible Gases

227

increase on the burned gas side while, of course, the tangential component remains the same. A consequence of the tangential velocity is that fluid elements in the oblique flame surface move along this surface. If the surface is curved, adjacent points traveling along the flame surface may move either farther apart (flame stretch) or closer together (flame compression). An oblique flame is curved if the velocity U of the approach flow varies in a direction y perpendicular to the direction of the approach flow. Strehlow showed that the quantity K1  (δL /U )(∂U/∂y) which is known as the Karlovitz flame stretch factor, is approximately equal to the ratio of the flame thickness δL to the flame curvature. The Karlovitz school has argued that excessive stretching can lead to local quenching of the reaction. Klimov [50], and later Williams [51], analyzed the propagation of a laminar flame in a shear flow with velocity gradient in terms of a more general stretch factor K 2  (δL /SL )(1/Λ) d Λ/dt where Λ is the area of an element of flame surface, dΛ/dt is its rate of increase, and δL/SL is a measure of the transit time of the gases passing through the flame. Stretch (K2 0) is found to reduce the flame thickness and to increase reactant consumption per unit area of the flame and large stretch (K2 0) may lead to extinction. On the other hand, compression (K2 0) increases flame thickness and reduces reactant consumption per unit incoming reactant area. These findings are relevant to laminar flamelets in a turbulent flame structure. Since the concern here is with the destruction of a contiguous laminar flame in a turbulent field, consideration must also be given to certain inherent instabilities in laminar flames themselves. There is a fundamental hydrodynamic instability as well as an instability arising from the fact that mass and heat can diffuse at different rates; that is, the Lewis number (Le) is nonunity. In the latter mechanism, a flame instability can occur when the Le number (α/D) is less than 1. Consider initially the hydrodynamic instability—that is, the one due to the flow—first described by Darrieus [52], Landau [53], and Markstein [54]. If no wrinkle occurs in a laminar flame, the flame speed SL is equal to the upstream unburned gas velocity U0. But if a minor wrinkle occurs in a laminar flame, the approach flow streamlines will either diverge or converge as shown in Fig. 4.45. Considering the two middle streamlines, one notes that, because of the curvature due to the wrinkle, the normal component of the velocity, with respect to the flame, is less than U0. Thus, the streamlines diverge as they enter the wrinkled flame front. Since there must be continuity of mass between

228

FIGURE 4.45 flame.

Combustion

Convergence–divergence of the flow streamlines due to a wrinkle in a laminar

the streamlines, the unburned gas velocity at the front must decrease owing to the increase of area. Since SL is now greater than the velocity of unburned approaching gas, the flame moves farther downstream and the wrinkle is accentuated. For similar reasons, between another pair of streamlines if the unburned gas velocity increases near the flame front, the flame bows in the upstream direction. It is not clear why these instabilities do not keep growing. Some have attributed the growth limit to nonlinear effects that arise in hydrodynamics. When the Lewis number is nonunity, the mass diffusivity can be greater than the thermal diffusivity. This discrepancy in diffusivities is important with respect to the reactant that limits the reaction. Ignoring the hydrodynamic instability, consider again the condition between a pair of streamlines entering a wrinkle in a laminar flame. This time, however, look more closely at the flame structure that these streamlines encompass, noting that the limiting reactant will diffuse into the flame zone faster than heat can diffuse from the flame zone into the unburned mixture. Thus, the flame temperature rises, the flame speed increases, and the flame wrinkles bow further in the downstream direction. The result is a flame that looks very much like the flame depicted for the hydrodynamic instability in Fig. 4.45. The flame surface breaks up continuously into new cells in a chaotic manner, as photographed by Markstein [54]. There appears to be, however, a higher-order stabilizing effect. The fact that the phenomenon is controlled by a limiting reactant means that this cellular condition can occur when the unburned premixed gas mixture is either fuelrich or fuel-lean. It should not be surprising, then, that the most susceptible mixture would be a lean hydrogen–air system. Earlier it was stated that the structure of a turbulent velocity field may be presented in terms of two parameters—the scale and the intensity of turbulence. The intensity was defined as the square root of the turbulent kinetic energy, which essentially gives a root-mean-square velocity fluctuation U. Three length scales were defined: the integral scale l0, which characterizes

229

Flame Phenomena in Premixed Combustible Gases

10 0. 1

.3δ L

0

0  L

3δ L

0δ L

ᐉk

1

10

10

0

00

0δ L

a



ᐉk

3

D

L



D

ᐉ /δ

a



10

D

a



10

D

,0

a

00



10

ᐉ k

ᐉ/ᐉk  100

ᐉ/ᐉk  10

U 1.0 SL

 δL

ᐉ/δ

ᐉk

ᐉ/ᐉk  1000

1

10

1 



L

a D

ᐉ/δ

ᐉ/δ

L



10 00

D a



ᐉk

0.1 1.0 1.0

10

Rᐍk Rλ

1.0

100

Rᐍ

10 100 10,000

FIGURE 4.46 Characteristic parametric relationships of premixed turbulent combustion. The Klimov–Williams criterion is satisfied below the heavy line lk  δL.

the large eddies; the Taylor microscale λ, which is obtained from the rate of strain; and the Kolmogorov microscale lk, which typifies the smallest dissipative eddies. These length scales and the intensity can be combined to form not one, but three turbulent Reynolds numbers: Rλ  Ul0/ν, Rλ  Uλ/ν, and Rk  Ulk/ν. From the relationship between l0, λ, and lk previously derived it is found that Rl ≈ Rλ2 ≈ Rk4 . There is now sufficient information to relate the Damkohler number Da and the length ratios l0/δL, lk/δL and l0/lk to a nondimensional velocity ratio U/SL and the three turbulence Reynolds numbers. The complex relationships are given in Fig. 4.46 and are very informative. The right-hand side of the figure has Rλ 100 and ensures the length-scale separation that is characteristic of high Reynolds number behavior. The largest Damkohler numbers are found in the bottom right corner of the figure. Using this graph and the relationship it contains, one can now address the question of whether and under what conditions a laminar flame can exist in a turbulent flow. As before, if allowance is made for flame front curvature effects, a laminar flame can be considered stable to a disturbance of sufficiently short wavelength; however, intense shear can lead to extinction. From solutions of the laminar flame equations in an imposed shear flow, Klimov [50] and Williams [51] showed that a conventional propagating flame may exist

230

Combustion

only if the stretch factor K2 is less than a critical value of unity. Modeling the area change term in the stretch expression as (1/Λ) d Λ/dt ≈ U/λ and recalling that δL ≈ ν /SL one can define the Karlovitz number for stretch in turbulent flames as K2 ≈

δLU SL Λ

with no possibility of negative stretch. Thus K2 ≈

δL U δL2 U δL2 U 1/ 2 l0 δ2   Rl  2L Rl3 / 2 SL λ ν λ ν l0 l0 l0

But as shown earlier lk  l0 /Rl3 / 4

or

l02  lk2 /Rl3 / 2

so that K 2  δL2 /lk2  (δL /lk )2 Thus, the criterion to be satisfied if a laminar flame is to exist in a turbulent flow is that the laminar flame thickness δL be less than the Kolmogorov microscale lk of the turbulence. The heavy line in Fig. 4.46 indicates the conditions δL  lk. This line is drawn in this fashion since δ U ν ν U U (U)2 1 K2 ≈ L ≈ 2 ≈ 2 ≈ ≈1 SL λ SL λ SL λ U  SL2 Rλ

or

⎛ U ⎞⎟2 ⎜⎜ ⎟ ≈ R λ ⎜⎜⎝ S ⎟⎟⎠ L

Thus for (U/SL)  1, Rλ  1; and for (U/SL)  10, Rλ  100. The other Reynolds numbers follow from Rk4  R2  Rl . Below and to the right of this line, the Klimov–Williams criterion is satisfied and wrinkled laminar flames may occur. The figure shows that this region includes both large and small values of turbulence Reynolds numbers and velocity ratios (U/SL) both greater and less than 1, but predominantly large Da. Above and to the left of the criterion line is the region in which lk δL. According to the Klimov–Williams criterion, the turbulent velocity gradients in this region, or perhaps in a region defined with respect to any of the characteristic lengths, are sufficiently intense that they may destroy a laminar flame.

Flame Phenomena in Premixed Combustible Gases

231

The figure shows U  SL in this region and Da is predominantly small. At the highest Reynolds numbers the region is entered only for very intense turbulence, U  SL. The region has been considered a distributed reaction zone in which reactants and products are somewhat uniformly dispersed throughout the flame front. Reactions are still fast everywhere, so that unburned mixture near the burned gas side of the flame is completely burned before it leaves what would be considered the flame front. An instantaneous temperature measurement in this flame would yield a normal probability density function—more importantly, one that is not bimodal.

3. The Turbulent Flame Speed Although a laminar flame speed SL is a physicochemical and chemical kinetic property of the unburned gas mixture that can be assigned, a turbulent flame speed ST is, in reality, a mass consumption rate per unit area divided by the unburned gas mixture density. Thus, ST must depend on the properties of the turbulent field in which it exists and the method by which the flame is stabilized. Of course, difficulty arises with this definition of ST because the time-averaged turbulent flame is bushy (thick) and there is a large difference between the area on the unburned gas side of the flame and that on the burned gas side. Nevertheless, many experimental data points are reported as ST. In his attempts to analyze the early experimental data, Damkohler [55] considered that large-scale, low-intensity turbulence simply distorts the laminar flame while the transport properties remain the same; thus, the laminar flame structure would not be affected. Essentially, his concept covered the range of the wrinkled and severely wrinkled flame cases defined earlier. Whereas a planar laminar flame would appear as a simple Bunsen cone, that cone is distorted by turbulence as shown in Fig. 4.43. It is apparent then, that the area of the laminar flame will increase due to a turbulent field. Thus, Damkohler [55] proposed for large-scale, small-intensity turbulence that (ST /SL )  (AL /AT ) where AL is the total area of laminar surface contained within an area of turbulent flame whose time-averaged area is AT. Damkohler further proposed that the area ratio could be approximated by (AL /AT )  1  (U 0 /SL ) which leads to the results ST  SL  U0

or

(ST /SL )  (U0 /SL ) 1

where U0 is the turbulent intensity of the unburned gases ahead of the turbulent flame front.

232

Combustion

Many groups of experimental data have been evaluated by semiempirical correlations of the type (ST /SL )  A(U0 /SL )  B and ST  A Re  B The first expression here is very similar to the Damkohler result for A and B equal to 1. Since the turbulent exchange coefficient (eddy diffusivity) correlates well with l0U for tube flow and, indeed, l0 is essentially constant for the tube flow characteristically used for turbulent premixed flame studies, it follows that U ∼ ε ∼ Re where Re is the tube Reynolds number. Thus, the latter expression has the same form as the Damkohler result except that the constants would have to equal 1 and SL, respectively, for similarity. Schelkin [56] also considered large-scale, small-intensity turbulence. He assumed that flame surfaces distort into cones with bases proportional to the square of the average eddy diameter (i.e., proportional to l0). The height of the cone was assumed proportional to U and to the time t during which an element of the wave is associated with an eddy. Thus, time can be taken as equal to (l0/SL). Schelkin then proposed that the ratio of ST/SL (average) equals the ratio of the average cone area to the cone base. From the geometry thus defined 1/ 2 AC  AB ⎡⎢⎣1  (4h2 /l02 ) ⎤⎥⎦

where AC is the surface area of the cone and AB is the area of the base. Therefore, 1/ 2 ST  SL ⎡⎢⎣1  (2U/SL )2 ⎤⎥⎦

For large values of (U/SL), that is, high-intensity turbulence, the preceding expression reduces to that developed by Damkohler: ST  U. A more rigorous development of wrinkled turbulent flames led Clavin and Williams [57] to the following result where isotropic turbulence is assumed: (ST /SL ) ∼ {1  [(U)2 /SL2 ]}1/ 2

233

Flame Phenomena in Premixed Combustible Gases

This result differs from Schelkin’s heuristic approach only by the factor of 2 in the second term. The Clavin-Williams expression is essentially restricted to the case of (U/SL)

1. For small (U/SL), the Clavin–Williams expression simplifies to (ST /SL ) ∼ 1  21 [(U)2 /SL ] which is quite similar to the Damkohler result. Kerstein and Ashurst [58], in a reinterpretation of the physical picture of Clavin and Williams, proposed the expression (ST /SL ) ∼ {1  (U/SL )2 }1/ 2 Using a direct numerical simulation, Yakhot [59] proposed the relation (ST /SL )  exp[(U/SL )2 /(ST /SL )2 ] For small-scale, high-intensity turbulence, Damkohler reasoned that the transport properties of the flame are altered from laminar kinetic theory viscosity ν0 to the turbulent exchange coefficient ε so that (ST /SL )  (ε /ν )1/ 2 This expression derives from SL  α1/2  ν1/2. Then, realizing that ε  Ul0, 1/ 2 ST ∼ SL ⎡⎣1  (λT /λL ) ⎤⎦

Schelkin [56] also extended Damkohler’s model by starting from the fact that the transport in a turbulent flame could be made up of molecular movements (laminar λL) and turbulent movements, so that 1/ 2 ST ∼ ⎡⎣ (λL  λT )/τ c ⎤⎦ ∼ {(λL /τ c ) ⎡⎣1  (λT /λL ) ⎤⎦ }

1/ 2

where the expression is again analogous to that for SL. (Note that λ is the thermal conductivity in this equation, not the Taylor scale.) Then it would follow that (λL /τ c ) ∼ SL

234

Turbulent burning velocity (ST/SL)

Combustion

30 25 20 Experiment (ReL  1000)

15 10 5 0

0

10 20 30 40 Turbulence intensity (U/SL)

50

FIGURE 4.47 General trend of experimental turbulent burning velocity (ST/SL) data as a function of turbulent intensity (U/SL) for Rl  1000 (from Ronney [39]).

and 1/ 2 ST ∼ SL ⎡⎣1  (λT /λL ) ⎤⎦

or essentially 1/ 2 (ST /SL ) ∼ ⎡⎣1  (ν T /ν L ) ⎤⎦

The Damkohler turbulent exchange coefficient ε is the same as νT, so that both expressions are similar, particularly in that for high-intensity turbulence ε ν. The Damkohler result for small-scale, high-intensity turbulence that (ST /SL ) ∼ Re1/ 2 is significant, for it reveals that (ST/SL) is independent of (U/SL) at fixed Re. Thus, as stated earlier, increasing turbulence levels beyond a certain value increases ST very little, if at all. In this regard, it is well to note that Ronney [39] reports smoothed experimental data from Bradley [46] in the form (ST/SL) versus (U/SL) for Re  1000. Ronney’s correlation of these data are reinterpreted in Fig. 4.47. Recall that all the expressions for small-intensity, largescale turbulence were developed for small values of (U/SL) and reported a linear relationship between (U/SL) and (ST/SL). It is not surprising, then, that a plot of these expressions—and even some more advanced efforts which also show linear relations—do not correlate well with the curve in Fig. 4.47. Furthermore, most developments do not take into account the effect of stretch on the turbulent flame. Indeed, the expressions reported here hold and show reasonable agreement with experiment only for (U/SL)

1. Bradley [46]

235

Flame Phenomena in Premixed Combustible Gases

102

10

Well-stirred reactor (thick flame, distributed combustion)

U

τc

SL

δ

τc 

τ K;

1/2

) (δ L/S L

(ν/ε)

Corregated flamelets (wrinkled flame with pockets) u/SL)  1

1

ᐉ

Laminar flame (thick flame)

0.1 0.1



;( τᐉ

) /LS L

Distributed reaction zone (thick-wrinkled flame, ) /u (ᐉ with possible extinctions)

K /δ L

(u/SL)  1 or R

e



1

Flamelet regime

Wrinkled flamelets (wrinkled flame) 1

10

102

103

ᐉ0 /δL FIGURE 4.48 Turbulent combustion regimes (from Abdel-Gayed et al. [61])

suggests that burning velocity is reduced with stretch, that is, as the Karlovitz stretch factor K2 increases. There are also some Lewis number effects. Refer to Ref. [39] for more details and further insights. The general data representation in Fig. 4.47 does show a rapid rise of (ST/SL) for values of (U/SL)

1. It is apparent from the discussion to this point that as (U/SL) becomes greater than 1, the character of the turbulent flame varies; and under the appropriate turbulent variables, it can change as depicted in Fig. 4.48, which essentially comes from Borghi [60] as presented by Abdel-Gayed et al. [61].

G. STIRRED REACTOR THEORY In the discussion of premixed turbulent flames, the case of infinitely fast mixing of reactants and products was introduced. Generally this concept is referred to as a stirred reactor. Many investigators have employed stirred reactor theory not only to describe turbulent flame phenomena, but also to determine overall reaction kinetic rates [23] and to understand stabilization in high-velocity streams [62]. Stirred reactor theory is also important from a practical point of view because it predicts the maximum energy release rate possible in a fixed volume at a particular pressure. Consider a fixed volume V into which fuel and air are injected at a fixed total mass flow rate m and temperature T0. The fuel and air react in the volume and the injection of reactants and outflow of products (also equal to m ) are so oriented that within the volume there is instantaneous mixing of the unburned

236

Combustion

gases and the reaction products (burned gases). The reactor volume attains some steady temperature TR and pressure P. The temperature of the gases leaving the reactor is, thus, TR as well. The pressure differential between the reactor and the exit is generally considered to be small. The mass leaving the reactor contains the same concentrations as those within the reactor and thus contains products as well as fuel and air. Within the reactor there exists a certain concentration of fuel (F) and air (A), and also a fixed unburned mass fraction, ψ. Throughout the reactor volume, TR, P, (F), (A), and ψ are constant and fixed; that is, the reactor is so completely stirred that all elements are uniform everywhere. Figure 4.49 depicts the stirred reactor concept in a generalized manner. The stirred reactor may be compared to a plug flow reactor in which premixed fuel–air mixtures flow through the reaction tube. In this case, the unburned gases enter at temperature T0 and leave the reactor at the flame temperature Tf. The system is assumed to be adiabatic. Only completely burned products leave the reactor. This reactor is depicted in Fig. 4.50. The volume required to convert all the reactants to products for the plug flow reactor is greater than that for the stirred reactor. The final temperature is, of course, higher than the stirred reactor temperature. It is relatively straightforward to develop the controlling parameters of a stirred reactor process. If ψ is defined as the unburned mass fraction, it must follow that the fuel–air mass rate of burning R B is R B  m (1  ψ )

P

TR

Fuel and air

Products that contain some fuel and air V

m,T0

m,TR (F) (A)

FIGURE 4.49

ψ

Variables of a stirred reactor system of fixed volume.

Tf (F)

m

Tu FIGURE 4.50

m

(A)

Variables in plug flow reactor.

x

Tf ≥ TR

Flame Phenomena in Premixed Combustible Gases

237

and the rate of heat evolution q is q  qm (1  ψ ) where q is the heat of reaction per unit mass of reactants for the given fuel–air ratio. Assuming that the specific heat of the gases in the stirred reactor can be represented by some average quantity c p , one can write an energy balance as  (1  ψ )  mc  p (TR  T0 ) mq For the plug flow reactor or any similar adiabatic system, it is also possible to define an average specific heat that takes its explicit definition from c p ˜ q/(Tf  T0 ) To a very good approximation these two average specific heats can be assumed equal. Thus, it follows that (1  ψ )  (TR  T0 )/(Tf  T0 ),

ψ  (Tf  TR )/(Tf  T0 )

The mass burning rate is determined from the ordinary expression for chemical kinetic rates; that is, the fuel consumption rate is given by d (F )/dt  (F)(A)Z  e−E/RTR  (F)2 (A/F)Z  eE/RTR where (A/F) represents the air–fuel ratio. The concentration of the fuel can be written in terms of the total density and unburned mass fraction (F) 

(F) 1 ρψ  ρψ (A)  (F) (A/F)  1

which permits the rate expression to be written as ⎛A⎞ d (F) 1 ρ 2 ψ 2 ⎜⎜ ⎟⎟⎟ Z  eE/RTR  2 ⎜⎝ F ⎠ dt ⎡ (A/F)  1⎤ ⎣ ⎦ Now the great simplicity in stirred reactor theory is realizable. Since (F), (A), and TR are constant in the reactor, the rate of conversion is constant. It is now possible to represent the mass rate of burning in terms of the preceding chemical kinetic expression: m (1  ψ )  V

(A)  (F) d (F) (F) dt

238

Combustion

or ⎡⎛ A ⎞ ⎤ 1 m (1  ψ )  V ⎢⎜⎜ ⎟⎟⎟  1⎥ 2 ⎢⎜⎝ F ⎠ ⎥⎡ ⎣ ⎦ ⎣ (A/F)  1⎤⎦

⎛ A ⎞⎟ 2 2 ⎜⎜ ⎟ ρ ψ Z  eE/RTR ⎝⎜ F ⎟⎠

From the equation of state, by defining B

Z ⎡ (A/F)  1⎤ ⎣ ⎦

and substituting for (1  ψ) in the last rate expression, one obtains 2 ⎛ m ⎞⎟ ⎛ A ⎞⎟ ⎛⎜ P ⎞⎟ ⎡ (Tf  TR )2 ⎤ BeE/RTR ⎜⎜ ⎟  ⎜⎜ ⎟ ⎜ ⎥ ⎟ ⎢ ⎜⎝ V ⎟⎠ ⎜⎝ F ⎟⎠ ⎜⎜⎝ RTR ⎟⎟⎠ ⎢ Tf  T0 ⎥ TR  T0 ⎣ ⎦

By dividing through by P2, one observes that (m /VP 2 )  f (TR )  f (A/F) or (m /VP 2 )(TR  T0 )  f (TR )  f (A/F) which states that the heat release is also a function of TR. This derivation was made as if the overall order of the air–fuel reaction were 2. In reality, this order is found to be closer to 1.8. The development could have been carried out for arbitrary overall order n, which would give the result (m /VP n )(TR  T0 )  f (TR )  f (A/F) A plot of (m /VP 2 )(TR  T0 ) versus TR reveals a multivalued graph that exhibits a maximum as shown in Fig. 4.51. The part of the curve in Fig. 4.51 that approaches the value Tx asymptotically cannot exist physically since the mixture could not be ignited at temperatures this low. In fact, the major part of the curve, which is to the left of Topt, has no physical meaning. At fixed volume and pressure it is not possible for both the mass flow rate and temperature of the reactor to rise. The only stable region exists between Topt to Tf. Since it is not possible to mix some unburned gases with the product mixture and still obtain the adiabatic flame temperature, the reactor parameter must go to zero when TR  Tf. The value of TR, which gives the maximum value of the heat release, is obtained by maximizing the last equation. The result is TR,opt 

Tf 1  (2 RTf /E )

239

Flame Phenomena in Premixed Combustible Gases

m (TR  T0) VP 2

Max

TX

Topt Tf TR

FIGURE 4.51 Stirred reactor parameter (m /VP2)(TR – T0) as a function of reactor temperature TR.

For hydrocarbons, the activation energy falls within a range of 120–160 kJ/mol and the flame temperature in a range of 2000–3000 K. Thus, (TR,opt /Tf ) ∼ 0.75 Stirred reactor theory reveals a fixed maximum mass loading rate for a fixed reactor volume and pressure. Any attempts to overload the system will quench the reaction. It is also worth noting that stirred reactor analysis for both non-dilute and dilute systems does give the maximum overall energy release rate that is possible for a given fuel–oxidizer mixture in a fixed volume at a given pressure. Attempts have been made to determine chemical kinetic parameters from stirred reactor measurements. The usefulness of such measurements at high temperatures and for non-dilute fuel–oxidizer mixtures is limited. Such analyses are based on the assumption of complete instantaneous mixing, which is impossible to achieve experimentally. However, for dilute mixtures at low and intermediate temperatures where the Damkohler number (Da  τm/τc) is small, studies have been performed to investigate the behaviors of reaction mechanisms [63]. Using the notation of Chapter 2 Section H3, Eqs. (2.61) and (2.62) may be rewritten in terms of the individual species equations and the energy equation. The species equations are given by dm j dt

 V ω j MW j  m *j  m j

j  1, … , n

(4.76)

where mj is the mass of the jth species in the reactor (and reactor outlet), V is the volume of the reactor, ω j is the chemical production rate of species j,

240

Combustion

MWj is the molecular weight of the jth species, m *j is the mass flow rate of species j in the reactor inlet, and m j is the mass flow rate of species j in the reactor outlet. For a constant mass flow rate and mass in the reactor, Eq. (4.76) may be written in terms of species mass fractions, Yj, as dY j dt



ω j MW j ρ



m (Y j*  Y j ) ρV

j  1,..., n

(4.77)

At steady state (dYj/dt  0), Yj 

V ω j MW j m

 Y j*

j  1,...,n

(4.78)

The energy equation for an adiabatic system at steady state is simply n

n

j 1

j 1

m ∑ Y j* h*j (T * )  m ∑ Y j h j (T )  0 where T

h j (T )  h 0f , j 

∫

c pj dT

Tref

Equation (4.78) is a set of nonlinear algebraic equation and may be solved using various techniques [64]. Often the nonlinear differential Eq. (4.77) are solved to the steady-state condition in place of the algebraic equations using the stiff ordinary differential equation solvers described in Chapter 2 [65]. See Appendix I for more information on available numerical codes.

H. FLAME STABILIZATION IN HIGH-VELOCITY STREAMS The values of laminar flame speeds for hydrocarbon fuels in air are rarely greater than 45 cm/s. Hydrogen is unique in its flame velocity, which approaches 240 cm/s. If one could attribute a turbulent flame speed to hydrocarbon mixtures, it would be at most a few hundred centimeters per second. However, in many practical devices, such as ramjet and turbojet combustors in which high volumetric heat release rates are necessary, the flow velocities of the fuel–air mixture are of the order of 50 m/s. Furthermore, for such velocities, the boundary layers are too thin in comparison to the quenching distance for stabilization to occur by the same means as that obtained in Bunsen burners. Thus, some other means for stabilization is necessary. In practice, stabilization

241

Flame Phenomena in Premixed Combustible Gases

(a)

(b)

(c)

(d)

FIGURE 4.52 Stabilization methods for high-velocity streams: (a) vee gutter, (b) rodoz sphere, (c) step or sudden expansion, and (d) opposed jet (after Strehlow, “Combustion Fundamentals,” McGraw-Hill, New York, 1985).

is accomplished by causing some of the combustion products to recirculate, thereby continuously igniting the fuel mixture. Of course, continuous ignition could be obtained by inserting small pilot flames. Because pilot flames are an added inconvenience—and because they can blow themselves out—they are generally not used in fast flowing turbulent streams. Recirculation of combustion products can be obtained by several means: (1) by inserting solid obstacles in the stream, as in ramjet technology (bluffbody stabilization); (2) by directing part of the flow or one of the flow constituents, usually air, opposed or normal to the main stream, as in gas turbine combustion chambers (aerodynamic stabilization), or (3) by using a step in the wall enclosure (step stabilization), as in the so-called dump combustors. These modes of stabilization are depicted in Fig. 4.52. Complete reviews of flame stabilization of premixed turbulent gases appear in Refs. [66, 67]. Photographs of ramjet-type burners, which use rods as bluff obstacles, show that the regions behind the rods recirculate part of the flow that reacts to form hot combustion products; in fact, the wake region of the rod acts as a pilot flame. Nicholson and Fields [68] very graphically showed this effect by placing small aluminum particles in the flow (Fig. 4.53). The wake pilot condition initiates flame spread. The flame spread process for a fully developed turbulent wake has been depicted [66], as shown in Fig. 4.54. The theory of flame spread in a uniform laminar flow downstream from a laminar mixing zone has been fully developed [12, 66] and reveals that the angle of flame spread is sin1(SL/U), where U is the main stream flow velocity. For a turbulent flame one approximates the spread angle by replacing SL by an appropriate turbulent flame speed ST. The limitations in defining ST in this regard were described in Section F. The types of obstacles used in stabilization of flames in high-speed flows could be rods, vee gutters, toroids, disks, strips, etc. But in choosing the

242

Combustion

FIGURE 4.53 Flow past a 0.5-cm rod at a velocity of 50 ft/s as depicted by an aluminum powder technique. Solid lines are experimental flow streamlines [59].

243

Flame Phenomena in Premixed Combustible Gases

Spreading of flame front

Uniform velocity profile far upstream

Percentage of Turbulent heat and mass transfer across combustion completed (schematic) boundary

Combustion nearly 100% completed Disk

20 40 60 80 100

Recirculation zone

Reverse flow of burnt gases

fer rans eat t h t n ule Turb Downstream velocity profile Turb ulen t hea t tran sfer

Turbulent Mass transfer to reaction zone

FIGURE 4.54 Recirculation zone and flame-spreading region for a fully developed turbulent wake behind a bluff body (after Williams [57]).

FIGURE 4.55 Flame-spreading interaction behind multiple bluff-body flame stabilizers.

bluff-body stabilizer, the designer must consider not only the maximum blowoff velocity the obstacle will permit for a given flow, but also pressure drop, cost, ease of manufacture, etc. Since the combustion chamber should be of minimum length, it is rare that a single rod, toroid, etc., is used. In Fig. 4.55, a schematic of flame spreading from multiple flame holders is given. One can readily see that multiple units can appreciably shorten the length of the combustion chamber. However, flame holders cause a stagnation pressure loss across the burner, and this pressure loss must be added to the large pressure drop due to burning. Thus, there is an optimum between the number of flame holders and pressure drop. It is sometimes difficult to use aerodynamic stabilization when large chambers are involved because the flow creating the recirculation would have to penetrate too far across the main stream. Bluff-body stabilization is not used in gas turbine systems because of the required combustor shape and the short lengths. In gas turbines a high weight penalty is paid for even the slightest increase

244

UBO (ft/s)

Combustion

400

0.5 in. 0.062 in.

200

3.5

0.016 in. 6.5 13 Air–fuel ratio

FIGURE 4.56 Blowoff velocities for various rod diameters as a function of air–fuel ratio. Short duct using premixed fuel–air mixtures. The 0.5 data are limited by chocking of duct (after Scurlock [60]).

in length. Because of reduced pressure losses, step stabilization has at times commanded attention. A wall heating problem associated with this technique would appear solvable by some transpiration cooling. In either case, bluff body or aerodynamic, blowout is the primary concern. In ramjets, the smallest frontal dimension for the highest flow velocity to be used is desirable; in turbojets, it is the smallest volume of the primary recirculation zone that is of concern; and in dump combustors, it is the least severe step. There were many early experimental investigations of bluff-body stabilization. Most of this work [69] used premixed gaseous fuel–air systems and typically plotted the blowoff velocity as a function of the air–fuel ratio for various stabilized sizes, as shown in Fig. 4.56. Early attempts to correlate the data appeared to indicate that the dimensional dependence of blowoff velocity was different for different bluff-body shapes. Later, it was shown that the Reynolds number range was different for different experiments and that a simple independent dimensional dependence did not exist. Furthermore, the state of turbulence, the temperature of the stabilizer, incoming mixture temperature, etc., also had secondary effects. All these facts suggest that fluid mechanics plays a significant role in the process. Considering that fluid mechanics does play an important role, it is worth examining the cold flow field behind a bluff body (rod) in the region called the wake. Figure 4.57 depicts the various stages in the development of the wake as the Reynolds number of the flow is increased. In region (1), there is only a slight slowing behind the rod and a very slight region of separation. In region (2), eddies start to form and the stagnation points are as indicated. As the Reynolds number increases, the eddy (vortex) size increases and the downstream stagnation point moves farther away from the rod. In region (3), the eddies become unstable, shed alternately, as shown in the figure, and (h/a)  0.3. As the velocity u increases, the frequency N of shedding increases; N  0.3(u/d). In region (4), there is a complete turbulent wake behind the

245

Flame Phenomena in Premixed Combustible Gases

(1) Re 10

(2) 20 Re 50

(3) 102 Re 105 h a 80 (4) Re ∼ 105

FIGURE 4.57 Flow fields behind rods as a function of Reynolds number.

body. The stagnation point must pass 90° to about 80° and the boundary layer is also turbulent. The turbulent wake behind the body is eventually destroyed downstream by jet mixing. The flow fields described in Fig. 4.57 are very specific in that they apply to cold flow over a cylindrical body. When spheres are immersed in a flow, region (3) does not exist. More striking, however, is the fact that when combustion exists over this Reynolds number range of practical interest, the shedding eddies disappear and a well-defined, steady vortex is established. The reason for this change in flow pattern between cold flow and a combustion situation is believed to be due to the increase in kinematic viscosity caused by the rise in temperature. Thus, the Reynolds number affecting the wake is drastically reduced, as discussed in the section of premixed turbulent flow. Then, it would be expected that in region (2) the Reynolds number range would be 10 Re 105. Flame holding studies by Zukoski and Marble [70, 71] showed that the ratio of the length of the wake (recirculation zone) to the diameter of the cylindrical flame holder was independent of the approach flow Reynolds number above a critical value of about 104. These Reynolds numbers are based

246

Combustion

on the critical dimension of the bluff body; that is, the diameter of the cylinder. Thus, one may assume that for an approach flow Reynolds number greater than 104, a fully developed turbulent wake would exist during combustion. Experiments [66] have shown that any original ignition source located upstream, near or at the flame holder, appears to establish a steady ignition position from which a flame spreads from the wake region immediately behind the stabilizer. This ignition position is created by the recirculation zone that contains hot combustion products near the adiabatic flame temperature [62]. The hot combustion products cause ignition by transferring heat across the mixing layer between the free-stream gases and the recirculation wake. Based on these physical concepts, two early theories were developed that correlated the existing data well. One was proposed by Spalding [72] and the other, by Zukoski and Marble [70, 71]. Another early theory of flame stabilization was proposed by Longwell et al. [73], who considered the wake behind the bluff body to be a stirred reactor zone. Considering the wake of a flame holder as a stirred reactor may be inconsistent with experimental data. It has been shown [66] that as blowoff is approached, the temperature of the recirculating gases remains essentially constant; furthermore, their composition is practically all products. Both of these observations are contrary to what is expected from stirred reactor theory. Conceivably, the primary zone of a gas turbine combustor might approach a state that could be considered completely stirred. Nevertheless, as will be shown, all three theories give essentially the same correlation. Zukoski and Marble [70, 71] held that the wake of a flame holder establishes a critical ignition time. Their experiments, as indicated earlier, established that the length of the recirculating zone was determined by the characteristic dimension of the stabilizer. At the blowoff condition, they assumed that the free-stream combustible mixture flowing past the stabilizer had a contact time equal to the ignition time associated with the mixture; that is, τw  τi, where τw is the flow contact time with the wake and τi is the ignition time. Since the flow contact time is given by τ w  L/U BO where L is the length of the recirculating wake and UBO is the velocity at blowoff, they essentially postulated that blowoff occurs when the Damkohler number has the critical value of 1; that is Da  (L/U BO )(1/τ i )  1 The length of the wake is proportional to the characteristic dimension of the stabilizer, the diameter d in the case of a rod, so that τ w ∼ (d/U BO )

Flame Phenomena in Premixed Combustible Gases

247

Thus it must follow that (U BO /d ) ∼ (1/τ i ) For second-order reactions, the ignition time is inversely proportional to the pressure. Writing the relation between pressure and time by referencing them to a standard pressure P0 and time τ0, one has (τ 0 /τ i )  (P/P0 ) where P is the actual pressure in the system of concern. The ignition time is a function of the combustion (recirculating) zone temperature, which, in turn, is a function of the air–fuel ratio (A/F). Thus, (U BO /dP ) ∼ (1/τ 0 P0 )  f (T )  f (A/F) Spalding [72] considered the wake region as one of steady-state heat transfer with chemical reaction. The energy equation with chemical reaction was developed and nondimensionalized. The solution for the temperature profile along the outer edge of the wake zone, which essentially heats the free stream through a mixing layer, was found to be a function of two nondimensional parameters that are functions of each other. Extinction or blowout was considered to exist when these dimensionless groups were not of the same order. Thus, the functional extinction condition could be written as (U BO d/α )  f (Z P n1d 2 /α ) where d is, again, the critical dimension, α is the thermal diffusivity, Z is the pre-exponential in the Arrhenius rate constant, and n is the overall reaction order. From laminar flame theory, the relationship SL ∼ (α RR)1/ 2 was obtained, so that the preceding expression could be modified by the relation SL ∼ (α Z P n1 )1/ 2 Since the final correlations have been written in terms of the air–fuel ratio, which also specifies the temperature, the temperature dependences were omitted. Thus, a new proportionality could be written as (SL2 /α ) ∼ Z P n1 (Z P n1d 2 /α )  (SL2 d 2 /α 2 )

248

Combustion

and the original functional relation would then be (U BO d/α ) ∼ f (SL d/α ) ∼ (U BO d/ν ) Both correlating parameters are in the form of Peclet numbers, and the air– fuel ratio dependence is in SL. Figure 4.58 shows the excellent correlation of data by the above expression developed from the Spalding analysis. Indeed, the power dependence of d with respect to blowoff velocity can be developed from the slopes of the lines in Fig. 4.58. Notice that the slope is 2 for values 106 Longwell; axial cylinders, naphtha  air De Zubay; discs, propane  air Baddour and Carr; spheres, propane  air Scurlock; rods, city gas  air Scurlock; rods, propane  air Baddour and Carr; rods, propane  air Scurlock; gutters, city gas  air

VBOd/α  (Peclet number based on VBO)

105 Slope 2

104

Two-Dimensional stabilizers

Slope 1.5

Three-Dimensional stabilizers

Slope 1.5

103

102

1

10 100 Sud/α  (Peclet number based on Su)

1000

FIGURE 4.58 Correlation of various blowoff velocity data by Spalding [63]; VBO  UBO, Su  SL.

249

Flame Phenomena in Premixed Combustible Gases

(UBOd/α) 104, which was found experimentally to be the range in which a fully developed turbulent wake exists. The correlation in this region should be compared to the correlation developed from the work of Zukoski and Marble. Stirred reactor theory was initially applied to stabilization in gas turbine combustor cans in which the primary zone was treated as a completely stirred region. This theory has sometimes been extended to bluff-body stabilization, even though aspects of the theory appear inconsistent with experimental measurements made in the wake of a flame holder. Nevertheless, it would appear that stirred reactor theory gives the same functional dependence as the other correlations developed. In the previous section, it was found from stirred reactor considerations that (m /VP 2 ) ≅ f (A/F) for second-order reactions. If m is considered to be the mass entering the wake and V its volume, then the following proportionalities can be written: m  ρ AU ∼ Pd 2U BO ,

V ∼ d3

where A is an area. Substituting these proportionalities in the stirred reactor result, one obtains [(Pd 2U BO )/(d 3 P 2 )]  f (A/F)  (U BO /dP ) which is the same result as that obtained by Zukoski and Marble. Indeed, in the turbulent regime, Spaldings development also gives the same form since in this regime the correlation can be written as the equality (U BO d/α )  const(SL d/α )2 Then it follows that (U BO /d ) ∼ (SL2 /α ) ∼ P n1 f (T )

or

(U BO /dP n1 ) ∼ f (T )

Thus, for a second-order reaction (U BO /dP ) ∼ f (T ) ∼ f (A/F) From these correlations it would be natural to expect that the maximum blowoff velocity as a function of air–fuel ratio would occur at the stoichiometric mixture ratio. For premixed gaseous fuel–air systems, the maxima do occur at this mixture ratio, as shown in Fig. 4.56. However, in real systems liquid fuels are injected upstream of the bluff-body flame holder in order to allow for mixing. Results [60] for such liquid injection systems show that the maximum

250

Combustion

blowoff velocity is obtained on the fuel-lean side of stoichiometric. This trend is readily explained by the fact that liquid droplets impinge on the stabilizer and enrich the wake. Thus, a stoichiometric wake undoubtedly occurs for a lean upstream liquid–fuel injection system. That the wake can be modified to alter blowoff characteristics was proved experimentally by Fetting et al. [74]. The trends of these experiments can be explained by the correlations developed in this section. When designed to have sharp leading edges, recesses in combustor walls cause flow separation, as shown in Fig. 4.52c. During combustion, the separated regions establish recirculation zones of hot combustion products much like the wake of bluff-body stabilizers. Studies [75] of turbulent propane–air mixtures stabilized by wall recesses in a rectangular duct showed stability limits significantly wider than that of a gutter bluff-body flame holder and lower pressure drops. The observed blowoff limits for a variety of symmetrically located wall recesses showed [66] substantially the same results, provided: (1) the recess was of sufficient depth to support an adequate amount of recirculating gas, (2) the slope of the recess at the upstream end was sharp enough to produce separation, and (3) the geometric construction of the recess lip was such that flow oscillations were not induced. The criterion for blowoff from recesses is essentially the same as that developed for bluff bodies, and L is generally taken to be proportional to the height of the recess [75]. The length of the recess essentially serves the same function as the length of the bluff-body recirculation zone unless the length is large enough for flow attachment to occur within the recess, in which case the recirculation length depends on the depth of the recess [12]. This latter condition applies to the so-called dump combustor, in which a duct with a small cross-sectional area exhausts coaxially with a right-angle step into a duct with a larger cross-section. The recirculation zone forms at the step. Recess stabilization appears to have two major disadvantages. The first is due to the large increase in heat transfer in the step area, and the second to flame spread angles smaller than those obtained with bluff bodies. Smaller flame spread angles demand longer combustion chambers. Establishing a criterion for blowoff during opposed-jet stabilization is difficult owing to the sensitivity of the recirculation region formed to its stoichiometry. This stoichiometry is well defined only if the main stream and opposed jet compositions are the same. Since the combustor pressure drop is of the same order as that found with bluff bodies [76], the utility of this means of stabilization is questionable.

I. COMBUSTION IN SMALL VOLUMES Combustion in small volumes has recently become of interest to applications of micropower generation [77–81], micropropulsion [82–94], microengines [95–102], microactuation [103–105], and microfuel reforming for fuel cells

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[106–108]. The push towards miniaturization of combustion based systems for micropower generation results in large part from the low specific energy of batteries and the high specific energy of liquid hydrocarbon fuels [77]. The interest in downsizing chemical thrusters, particularly solid propellant systems, results from the potential gain in the thrust-to-weight ratio, T/W  ᐍ1, since weight is proportional to the linear characteristic dimension of the system cubed (ᐍ3) and thrust is proportional to the length scale squared (T  PcAt  ᐍ2 where Pc is the combustion chamber pressure and At is the nozzle throat cross-sectional area). Since heat conduction rates in small-scale systems can be significant, microreactors benefit from controlled isothermal and safe operation eliminating the potential of thermal runaway. Small reactors are useful for working with small quantities of hazardous or toxic materials and can easily be scaled-up through the number of reactors versus the size of the reactor. In addition, the potential fabrication of small-scale devices using microelectromechanical systems (MEMS) or rapid prototyping techniques, which have favorable characteristics for mass production and low cost, has been a driver of the field. Fernandez-Pello provides a review of many of the issues important to reacting flows at the microscale [78]. Many of the issues important to combustion at the microscale are the same as those discussed earlier for the macroscale. Currently, characteristic length scales of microcombustion devices are larger than the mean free path of the gases flowing through the devices so that the physical–chemical behavior of the fluids is fundamentally the same as in their macroscale counterparts (i.e., the Knudsen number remains much less than unity and therefore, the fluid medium still behaves as a continuum and the no-slip condition applies at surfaces) [78, 109]. However, the small scales involved in microdevices emphasize particular characteristics of fluid mechanics, heat transfer, and combustion often not important in macroscale systems. For example, the effect of heat loss to the wall is generally insignificant and ignored in a macroscale device, but is a decisive factor in a microscale combustor. These characteristics may be identified by examining the nondimensional conservation equations of momentum, energy, and species [78]. From the nondimensional conservation equations, it is evident that, compared to macroscale systems, microcombustors operate at lower Re (Uᐍ/ν) and Pe (Uᐍ/α) numbers, where U is a characteristic flow velocity of the system and ν and α are the kinematic and thermal diffusivities. Consequently, viscous and diffusive effects play a greater role. The flows are less turbulent and laminar conditions generally prevail. Boundary conditions, which are usually not very influential in large-scale systems, play a more significant role at smaller scales thus amplifying the influence of interfacial phenomena. Diffusive processes, which in micron-sized channels can be fast, will largely dominate species mixing. However, they can be too slow to be effective in millimeter-sized combustion chambers, since the residence time (τr  ᐍ/U) may not be sufficient. The greater surface tension and viscous forces result in larger pumping requirements. To overcome some of these fluid mechanics issues, many microcombustors operate at higher pressures to achieve high Re numbers.

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As length scale is decreased, the velocity, temperature, and species gradients at boundaries would have to increase to preserve the bulk flow conditions of a macroscale system, further making it difficult to maintain large differences between the wall and bulk flow variables of a microscale device. Considering heat transfer, Biot numbers (hᐍ/ks) associated with microstructures will generally be much less than unity, resulting in nearly uniform body temperatures. The small buoyancy and the low thermal conductivity of air render heat losses from a combustor to the surrounding environment relatively small when compared to total heat generation making heat loss at the external surface often the rate limiting heat transfer process. Fourier numbers (αt/ᐍ2) indicate that thermal response times of structures are small and can approach the response times of the fluid flow. Because view factors increase with decreasing characteristic length, radiative heat transfer also plays a significant role in small-scale devices. Consequently, for microcombustors, the structures become intimately coupled to the flame and reaction dynamics. Chemical time scales (τc) are independent of the device length scale. As discussed in Chapter 2, τc depends on the reactant concentrations, the reacting temperature, and types of fuel and oxidizer, and because of Arrhenius kinetics, generally increases exponentially as the temperature decreases. The reaction temperature is strongly influenced by the increased surface-to-volume ratio and the shortened flow residence time as length scales are decreased. As the combustion volume decreases, the surface-to-volume ratio increases approximately inversely with critical dimension. Consequently, heat losses to the wall increase as well as the potential destruction of radical species, which are the two dominant mechanisms of flame quenching. Thermal management in microcombustors is critical and the concepts of quenching distance discussed earlier are directly applicable. Thermal quenching of confined flames occurs when the heat generated by the combustion process fails to keep pace with the heat loss to the walls. Combustor bodies, in most cases, act as a thermal sink. Consequently, as the surface-to-volume ratio increases, the portion of heat lost to the combustor body increases and less enthalpy is preserved in the combustion product, which further lowers the combustion temperature and slows kinetic rates, reducing heat generation and quenching the flame. The other interfacial phenomenon, radical quenching, removes transient species crucial to the propagation of the chain mechanism leading to extinction. In order to overcome these two interfacial phenomena, “excess enthalpy” combustors [110, 111] have been applied to small-scale devices, where the high-temperature combustion product gases are used to preheat the cold reactants without mass transfer. In the excess enthalpy combustor, thermal energy is transferred to the reactants so that the total reactant enthalpy in the combustor is higher than that in the incoming cold reactants. These burners have been demonstrated to significantly extend flammability limits at small scales [111]. The high surface-to-volume ratio and small length scales also favor the use of surface catalysis at the microscale

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[112, 113]. However, catalytic combustion deposits a portion of the liberated energy directly into the structure complicating thermal management. For efficient combustion, the residence time (τR  ᐍ/U) must be greater than the chemical time (τC). The ratio of these times is the Damkohler number (τR /τC). In general, τR will decrease with decreasing ᐍ and therefore to sustain combustion, chemical times will also need to be reduced. For efficient gasphase combustion, high inlet, wall, and combustion temperatures for increased kinetic rates, operation with stoichiometric mixtures, and the use of highly energetic fuels are all approaches to enhance combustion. For non-premixed combustion (Chapter 6), the flow residence time has to be larger than the combination of the mixing time scale and the chemical reaction time scale. A second Damkohler number based on the mixing time versus the chemical time is therefore important. For liquid fuels, evaporation times also need to be scaled with ᐍ if efficient combustion is to be achieved. For sprays, droplet sizes need to be reduced to shorten evaporation times (discussed in Chapter 6), which implies greater pressure and energy requirements for atomization. To address this aspect, microelectrospray atomizers have been considered [114, 115]. An alternative approach under investigation is to use film-cooling techniques as a means to introduce liquid fuels into the combustion chamber, since surface-tovolume ratios are high [116]. Other processes that have increased importance at small length scales such as thermal creep (transpiration) and electrokinetic effects are also being considered for use in microcombustors. For example, transpiration effects are currently being investigated by Ochoa et al. [117] to supply fuel to the combustion chamber creating an in-situ thermally driven reactant flow at the front end of the combustor. Several fundamental studies have been devoted to flame propagation in microchannels, emphasizing the effects of flame wall coupling on flame propagation. In these studies, heat conduction through the structure is observed to broaden the reaction zone [118]. However, heat losses to the environment decrease the broadening effect and eventually result in flame quenching. While the increase in flame thickness appears to be a drawback for designing high power density combustors because it suggests that proportionally more combustor volume is required, the increase in burning rate associated with preheating the reactants more than compensates for this effect and the net result is an increase in power density. It has also been shown that if wall temperatures are high enough, combustion in passages smaller than the quenching diameter is possible [119] and that streamwise heat conduction is as important as heat transfer perpendicular to the gas flow [111]. Many interesting flame bifurcations and instabilities have been observed in microchannels. In particular, non-monotonic dependencies of the flame speed on equivalence ratio, the existence of two flame transitions, a direct transition and an extinction transition, depending on channel width, Lewis number and flow velocity, and cellular flame structures have all been reported [120, 121].

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PROBLEMS (Those with an asterisk require a numerical solution.) 1. A stoichiometric fuel–air mixture flowing in a Bunsen burner forms a well-defined conical flame. The mixture is then made leaner. For the same flow velocity in the tube, how does the cone angle change for the leaner mixture? That is, does the cone angle become larger or smaller than the angle for the stoichiometric mixture? Explain. 2. Sketch a temperature profile that would exist in a one-dimensional laminar flame. Superimpose on this profile a relative plot of what the rate of energy release would be through the flame as well. Below the inflection point in the temperature profile, large amounts of HO2 are found. Explain why. If flame was due to a first-order, one-step decomposition reaction, could rate data be obtained directly from the existing temperature profile? 3. In which of the two cases would the laminar flame speed be greater: (1) oxygen in a large excess of a wet equimolar CO¶CO2 mixture or (2) oxygen in a large excess of a wet equimolar CO¶N2 mixture? Both cases are ignitable, contain the same amount of water, and have the same volumetric oxygen–fuel ratio. Discuss the reasons for the selection made. 4. A gas mixture is contained in a soap bubble and ignited by a spark in the center so that a spherical flame spreads radially through the mixture. It is assumed that the soap bubble can expand. The growth of the flame front along a radius is followed by some photographic means. Relate the velocity of the flame front as determined from the photographs to the laminar flame speed as defined in the text. If this method were used to measure flame speeds, what would be its advantages and disadvantages? 5. On what side of stoichiometric would you expect the maximum flame speed of hydrogen–air mixtures? Why? 6. A laminar flame propagates through a combustible mixture in a horizontal tube 3 cm in diameter. The tube is open at both ends. Due to buoyancy effects, the flame tilts at a 45° angle to the normal and is planar. Assume the tilt is a straight flame front. The normal laminar flame speed for the combustible mixture is 40 cm/s. If the unburned gas mixture has a density of 0.0015 gm/cm3, what is the mass burning rate of the mixture in grams per second under this laminar flow condition? 7. The flame speed for a combustible hydrocarbon–air mixture is known to be 30 cm/s. The activation energy of such hydrocarbon reactions is generally assumed to be 160 kJ/mol. The true adiabatic flame temperature for this mixture is known to be 1600 K. An inert diluent is added to the mixture to lower the flame temperature to 1450 K. Since the reaction is of second-order, the addition of the inert can be considered to have no other effect on any property of the system. Estimate the flame speed after the diluent is added. 8. A horizontal long tube 3 cm in diameter is filled with a mixture of methane and air in stoichiometric proportions at 1 atm and 27°C. The column

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is ignited at the left end and a flame propagates at uniform speed from left to right. At the left end of the tube is a convergent nozzle that has a 2-cm diameter opening. At the right end there is a similar nozzle 0.3 cm in diameter at the opening. Calculate the velocity of the flame with respect to the tube in centimeters per second. Assume the following: (a) The effect of pressure increase on the burning velocity can be neglected; similarly, the small temperature increase due to adiabatic compression has a negligible effect. (b) The entire flame surface consumes combustible gases at the same rate as an ideal one-dimensional flame. (c) The molecular weight of the burned gases equals that of the unburned gases. (d) The flame temperature is 2100 K. (e) The normal burning velocity at stoichiometric is 40 cm/s. Hint: Assume that the pressure in the burned gases is essentially 1 atm. In calculating the pressure in the cold gases make sure the value is correct to many decimal places. 9. A continuous flow stirred reactor operates off the decomposition of gaseous ethylene oxide fuel. If the fuel injection temperature is 300 K, the volume of the reactor is 1500 cm3, and the operating pressure is 20 atm, calculate the maximum rate of heat evolution possible in the reactor. Assume that the ethylene oxide follows homogeneous first-order reaction kinetics and that values of the reaction rate constant k are k  3.5 s1 at 980 K k  50 s1 at 1000 K k  600 s1 at 1150 K Develop any necessary rate data from these values. You are given that the adiabatic decomposition temperature of gaseous ethylene oxide is 1300 K. The heat of formation of gaseous ethylene oxide at 300 K is 50 kJ/mol. The overall reaction is C2 H 4 O → CH 4  CO 10. What are the essential physical processes that determine the flammability limit? 11. You want to measure the laminar flame speed at 273 K of a homogeneous gas mixture by the Bunsen burner tube method. If the mixture to be measured is 9% natural gas in air, what size would you make the tube diameter? Natural gas is mostly methane. The laminar flame speed of the mixture can be taken as 34 cm/s at 298 K. Other required data can be found in standard reference books.

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12. A ramjet has a flame stabilized in its combustion chamber by a single rod whose diameter is 1.25 cm. The mass flow of the unburned fuel–air mixture entering the combustion chamber is 22.5 kg/s, which is the limit amount that can be stabilized at a combustor pressure of 3 atm for the cylindrical configuration. The ramjet is redesigned to fly with the same fuel– air mixture and a mass flow rate twice the original mass flow in the same size (cross-section) combustor. The inlet diffuser is such that the temperature entering the combustor is the same as in the original case, but the pressure has dropped to 2 atm. What is the minimum size rod that will stabilize the flame under these new conditions? 13. A laminar flame propagates through a combustible mixture at 1 atm pressure, has a velocity of 50 cm/s and a mass burning rate of 0.1 g/s cm2. The overall chemical reaction rate is second-order in that it depends only on the fuel and oxygen concentrations. Now consider a turbulent flame propagating through the same combustible mixture at a pressure of 10 atm. In this situation the turbulent intensity is such that the turbulent diffusivity is 10 times the laminar diffusivity. Estimate the turbulent flame propagation and mass burning rates. 14. Discuss the difference between explosion limits and flammability limits. Why is the lean flammability limit the same for both air and oxygen? 15. Explain briefly why halogen compounds are effective in altering flammability limits. 16. Determine the effect of hydrogen addition on the laminar flame speed of a stoichiometric methane–air mixture. Vary the fuel mixture concentration from 100% CH4 to a mixture of 50% CH4 and 50% H2 in increments of 10% H2 maintaining a stoichiometric mixture. Plot the laminar flame speed as a function of percent H2 in the initial mixture and explain the trends. Using the temperature profile, determine how the flame thickness varies with H2 addition. The laminar flame speeds can be evaluated using the freely propagating laminar premixed code of CHEMKIN (or an equivalent code). A useful reaction mechanism for methane oxidation is GRIMech3.0 (developed from support by the Gas Research Institute) and can be downloaded from the website http://www.me.berkeley.edu/gri-mech/ version30/text30.html#thefiles. 17.* Calculate the laminar burning velocity as a function of pressure at 0.25, 1, and 3 atmospheres of a stoichiometric methane–air mixture. Discuss the results and compare the values to the experimental measurements in Table E3. The laminar premixed flame code of CHEMKIN (or an equivalent code) may be used with the Gas Research Institute reaction mechanism for methane oxidation (http://www.me.berkeley.edu/gri-mech/ version30/text30.html#thefiles). 18.*The primary zone of a gas turbine combustor is modeled as a perfectly stirred reactor. The volume of the primary zone is estimated to be 1.5  103 cm3. The combustor operates at a pressure of 10 atm with an

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air inlet temperature of 500 K. For a stoichiometric methane–air mixture, determine the minimum residence time (maximum flow rate) at which blowout occurs. Also determine the fuel-lean and fuel-rich mixture equivalence ratios at which blowout occurs for a reactor residence time equal to twice the time of that determined above. Compare these values to the lean and rich flammability limits given in Appendix E for a stoichiometric methane–air mixture. The perfectly stirred reactor codes of CHEMKIN (or an equivalent code) may be used with the Gas Research Institute reaction mechanism for methane oxidation (http://www.me.berkeley.edu/ gri-mech/verson30/text30.hml#thefiles).

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98. Lee, C. H., Jiang, K. C., Jin, P., and Prewett, P. D., Microelectron. Eng. 73–74, 529–534 (2004). 99. Dahm, W. J. A., Ni, J., Mijit, K., Mayor, R., Qiao, G., Benjamin, A., Gu, Y., Lei, Y., and Papke, M., Micro internal combustion swing engine (MICSE) for portable power generation systems. In the 40th AIAA Aerospace Sciences Meeting and Exhibit, Reno, 2002. 100. Lee, D. H., Park, D. E., Yoon, E., and Kwon, S., J. Heat Transfer-Trans. ASME 125, 487–493 (2003). 101. Aichlmayr, H. T., Kittelson, D. B., and Zachariah, M. R., Combust. Flame 135, 227–248 (2003). 102. Yang, W., Proc. Combust. Inst. 28 (2000). 103. Rossi, C., and Esteve, D., Sensors Actuators A Phys. 120, 297–310 (2005). 104. Hong, C. C., Murugesan, S., Kim, S., Beaucage, G., Choi, J. W., and Ahn, C. H., Lab on a Chip 3, 281–286 (2003). 105. DiBiaso, H. H., English, B. A., and Allen, M. G., Sensors Actuators A Phys. 111, 260–266 (2004). 106. Shah, K., Ouyang, X., and Besser, R. S., Chem. Eng. Technol. 28, 303–313 (2005). 107. Norton, A. G., Deshmukh, S. R., Wetzel, E. D., and Vlachos, D. G., Microreactor Technol. Process Intens. 914, 179–193 (2005). 108. Cao, C. S., Wang, Y., Holladay, J. D., Jones, E. O., and Palo, D. R., AICHE J. 51, 982–988 (2005). 109. Karniadakis, G. E., and Beskok, A., “Micro Flows.” Springer-Verlag, New York, 2002. 110. Vican, J., Gajdeczko, B. F., Dryer, F. L., Milius, D. L., Aksay, I. A., and Yetter, R. A., Proc. Combust. Inst. 29, 909 (2002). 111. Ronney, P. D., Combust. Flame 135, 421 (2003). 112. Maruta, K., Takeda, K., Ahn, T., Borer, K., Sitski, L., Ronney, P. D., and Deutschmann, O., Proc. Combust. Inst. 29, 957 (2002). 113. Norton, D. G., Wetzel, E. D., and Vlachos, D. G., Ind. Eng. Chem. Res. 45, 76 (2006). 114. Kyritsis, D. C., Guerrero-Arias, I., Roychoudhury, S., and Gomez, A., Proc. Combust. Inst. 29, 965–972 (2002). 115. Gemci, T. and Chigier, N., “Electrodynamic Atomization for MEMS Combustion Systems,” AIAA 2003-675 (2003). 116. Stanchi, S., Dunn-Rankin, D., Sirignano, W. A., “Combustor Miniaturization with Liquid Fuel Filming,” AIAA 2003-1163 (2003). 117. Ochoa, F., Eastwood, C., Ronney, P. D., and Dunn, B., (2003) Thermal Transpiration Based Microscale Propulsion and Power Generation Devices, 7th International Workshop on Microgravity Combustion and Chemically Reacting Systems, NASA/CP-2003-21376, 2003. 118. Leach, T. T., Cadou, C. P., and Jackson, G. S., Combust Theory Modelling 10, 1, 85 (2006). 119. Zamashchikov, V. V., Combust. Sci. Technol. 166, 1 (2001). 120. Ju, Y., and Xu, B., Combust. Sci. Technol. 178, 1723 (2006). 121. Prakash, S., Armijo, A. D., Masel, R. I., and Shannon, M. A., AIChE J. 53, 6, 1568 (2007).

Chapter 5

Detonation A. INTRODUCTION Established usage of certain terms related to combustion phenomena can be misleading, for what appear to be synonyms are not really so. Consequently, this chapter begins with a slight digression into the semantics of combustion, with some brief mention of subjects to be covered later.

1. Premixed and Diffusion Flames The previous chapter covered primarily laminar flame propagation. By inspecting the details of the flow, particularly high-speed or higher Reynolds number flow, it was possible to consider the subject of turbulent flame propagation. These subjects (laminar and turbulent flames) are concerned with gases in the premixed state only. The material presented is not generally adaptable to the consideration of the combustion of liquids and solids, or systems in which the gaseous reactants diffuse toward a common reacting front. Diffusion flames can best be described as the combustion state controlled by mixing phenomena—that is, the diffusion of fuel into oxidizer, or vice versa—until some flammable mixture ratio is reached. According to the flow state of the individual diffusing species, the situation may be either laminar or turbulent. It will be shown later that gaseous diffusion flames exist, that liquid burning proceeds by a diffusion mechanism, and that the combustion of solids and some solid propellants falls in this category as well.

2. Explosion, Deflagration, and Detonation Explosion is a term that corresponds to rapid heat release (or pressure rise). An explosive gas or gas mixture is one that will permit rapid energy release, as compared to most steady, low-temperature reactions. Certain gas mixtures (fuel and oxidizer) will not propagate a burning zone or combustion wave. These gas mixtures are said to be outside the flammability limits of the explosive gas. Depending upon whether the combustion wave is a deflagration or detonation, there are limits of flammability or detonation. In general, the combustion wave is considered as a deflagration only, although the detonation wave is another class of the combustion wave. The 261

262

Combustion

detonation wave is, in essence, a shock wave that is sustained by the energy of the chemical reaction in the highly compressed explosive medium existing in the wave. Thus, a deflagration is a subsonic wave sustained by a chemical reaction and a detonation is a supersonic wave sustained by chemical reaction. In the normal sense, it is common practice to call a combustion wave a “flame,” so combustion wave, flame, and deflagration have been used interchangeably. It is a very common error to confuse a pure explosion and a detonation. An explosion does not necessarily require the passage of a combustion wave through the exploding medium, whereas an explosive gas mixture must exist in order to have either a deflagration or a detonation. That is, both deflagrations and detonations require rapid energy release; but explosions, though they too require rapid energy release, do not require the presence of a waveform. The difference between deflagration and detonation is described qualitatively, but extensively, by Table 5.1 (from Friedman [1]).

3. The Onset of Detonation Depending upon various conditions, an explosive medium may support either a deflagration or a detonation wave. The most obvious conditions are confinement, mixture ratio, and ignition source. Original studies of gaseous detonations have shown no single sequence of events due primarily to what is now known as the complex cellular structure of a detonation wave. The primary result of an ordinary thermal initiation always appears to be a flame that propagates with subsonic speed. When conditions are such that the flame causes adiabatic compression of the still unreacted mixture ahead of it, the flame velocity increases. According to some early observations,

TABLE 5.1 Qualitative Deflagration in Gases

Differences

Between

Detonations

and

Usual magnitude of ratio Ratio

Detonation

Deflagration

uu/cua

5–10

ub/uu

0.4–0.7

4–16

Pb/Pu

13–55

0.98–0.976

Tb/Tu

8–21

4–16

ρb/ρu

1.4–2.6

0.06–0.25

a

cu is the acoustic velocity in the unburned gases. uu/cu is the Mach number of the wave.

0.0001–0.03

Detonation

263

the speed of the flame seems to rise gradually until it equals that of a detonation wave. Normally, a discontinuous change of velocity is observed from the low flame velocity to the high speed of detonation. In still other observations, the detonation wave has been seen to originate apparently spontaneously some distance ahead of the flame front. The place of origin appears to coincide with the location of a shock wave sent out by the expanding gases of the flame. Modern experiments and analysis have shown that these seemingly divergent observations were in part attributable to the mode of initiation. In detonation phenomena, one can consider that two modes of initiation exist: a slower mode, sometimes called thermal initiation, in which there is transition from deflagration; and a fast mode brought about by an ignition blast or strong shock wave. Some [2] refer to these modes as self-ignition and direct ignition, respectively. When an explosive gas mixture is placed in a tube having one or both ends open, a combustion wave can propagate when the tube is ignited at an open end. This wave attains a steady velocity and does not accelerate to a detonation wave. If the mixture is ignited at one end that is closed, a combustion wave is formed; and, if the tube is long enough, this wave can accelerate to a detonation. This thermal initiation mechanism is described as follows. The burned gas products from the initial deflagration have a specific volume of the order of 5–15 times that of the unburned gases ahead of the flame. Since each preceding compression wave that results from this expansion tends to heat the unburned gas mixture somewhat, the sound velocity increases and the succeeding waves catch up with the initial one. Furthermore, the preheating tends to increase the flame speed, which then accelerates the unburned gas mixture even further to a point where turbulence is developed in the unburned gases. Then, a still greater velocity and acceleration of the unburned gases and compression waves are obtained. This sequence of events forms a shock that is strong enough to ignite the gas mixture ahead of the front. The reaction zone behind the shock sends forth a continuous compression wave that keeps the shock front from decaying, and so a detonation is obtained. At the point of shock formation a detonation forms and propagates back into the unburned gases [2, 3]. Transverse vibrations associated with the onset of detonation have been noticed, and this observation has contributed to the understanding of the cellular structure of the detonation wave. Photographs of the onset of detonation have been taken by Oppenheim and co-workers [3] using a stroboscopic-laser-Schlieren technique. The reaction zone in a detonation wave is no different from that in other flames, in that it supplies the sustaining energy. A difference does exist in that the detonation front initiates chemical reaction by compression, by diffusion of both heat and species, and thus inherently maintains itself. A further, but not essential, difference worth noting is that the reaction takes place with extreme rapidity in highly compressed and preheated gases. The transition length for deflagration to detonation is of the order of a meter for highly reactive fuels such as acetylene, hydrogen, and ethylene, but larger for most other hydrocarbon–air mixtures. Consequently, most laboratory

264

Combustion

results for detonation are reported for acetylene and hydrogen. Obviously, this transition length is governed by many physical and chemical aspects of the experiments. Such elements as overall chemical composition, physical aspects of the detonation tube, and initiation ignition characteristics can all play a role. Interestingly, some question exists as to whether methane will detonate at all. According to Lee [2], direct initiation of a detonation can occur only when a strong shock wave is generated by a source and this shock retains a certain minimum strength for some required duration. Under these conditions “the blast and reaction front are always coupled in the form of a multiheaded detonation wave that starts at the (ignition) source and expands at about the detonation velocity” [2]. Because of the coupling phenomena necessary, it is apparent that reaction rates play a role in whether a detonation is established or not. Thus ignition energy is one of the dynamic detonation parameters discussed in the next section. However, no clear quantitative or qualitative analysis exists for determining this energy, so this aspect of the detonation problem will not be discussed further.

B. DETONATION PHENOMENA Scientific studies of detonation phenomena date back to the end of the nineteenth century and persist as an active field of investigation. A wealth of literature has developed over this period; consequently, no detailed reference list will be presented. For details and extensive references the reader should refer to books on detonation phenomena [4], Williams’ book on combustion [5], and the review by Lee [6]. Since the discussion of the detonation phenomena to be considered here will deal extensively with premixed combustible gases, it is appropriate to introduce much of the material by comparison with deflagration phenomena. As the data in Table 5.1 indicate, deflagration speeds are orders of magnitude less than those of detonation. The simple solution for laminar flame speeds given in Chapter 4 was essentially obtained by starting with the integrated conservation and state equations. However, by establishing the Hugoniot relations and developing a Hugoniot plot, it was shown that deflagration waves are in the very low Mach number regime; then it was assumed that the momentum equation degenerates and the situation through the wave is one of uniform pressure. The degeneration of the momentum equation ensures that the wave velocity to be obtained from the integrated equations used will be small. In order to obtain a deflagration solution, it was necessary to have some knowledge of the wave structure and the chemical reaction rates that affected this structure. As will be shown, the steady solution for the detonation velocity does not involve any knowledge of the structure of the wave. The Hugoniot plot discussed in Chapter 4 established that detonation is a large Mach number phenomenon. It is apparent, then, that the integrated momentum equation is included in obtaining a solution for the detonation velocity. However, it was also noted that there are four integrated conservation and state equations and five unknowns. Thus, other

Detonation

265

considerations were necessary to solve for the velocity. Concepts proposed by Chapman [7] and Jouguet [8] provided the additional insights that permitted the mathematical solution to the detonation velocity problem. The solution from the integrated conservation equations is obtained by assuming the detonation wave to be steady, planar, and one-dimensional; this approach is called Chapman– Jouguet (C–J) theory. Chapman and Jouguet established for these conditions that the flow behind the supersonic detonation is sonic. The point on the Hugoniot curve that represents this condition is called the C–J point and the other physical conditions of this state are called the C–J conditions. What is unusual about the C–J solution is that, unlike the deflagration problem, it requires no knowledge of the structure of the detonation wave and equilibrium thermodynamic calculations for the C–J state suffice. As will be shown, the detonation velocities that result from this theory agree very well with experimental observations, even in near-limit conditions when the flow structure near the flame front is highly threedimensional [6]. Reasonable models for the detonation wave structure have been presented by Zeldovich [9], von Neumann [10], and Döring [11]. Essentially, they constructed the detonation wave to be a planar shock followed by a reaction zone initiated after an induction delay. This structure, which is generally referred to as the ZND model, will be discussed further in a later section. As in consideration of deflagration phenomena, other parameters are of import in detonation research. These parameters—detonation limits, initiation energy, critical tube diameter, quenching diameter, and thickness of the supporting reaction zone—require a knowledge of the wave structure and hence of chemical reaction rates. Lee [6] refers to these parameters as “dynamic” to distinguish them from the equilibrium “static” detonation states, which permit the calculation of the detonation velocity by C–J theory. Calculation of the dynamic parameters using a ZND wave structure model do not agree with experimental measurements, mainly because the ZND structure is unstable and is never observed experimentally except under transient conditions. This disagreement is not surprising, as numerous experimental observations show that all self-sustained detonations have a three-dimensional cell structure that comes about because reacting blast “wavelets” collide with each other to form a series of waves which transverse to the direction of propagation. Currently, there are no suitable theories that define this three-dimensional cell structure. The next section deals with the calculation of the detonation velocity based on C–J theory. The subsequent section discusses the ZND model in detail, and the last deals with the dynamic detonation parameters.

C. HUGONIOT RELATIONS AND THE HYDRODYNAMIC THEORY OF DETONATIONS If one is to examine the approach to calculate the steady, planar, one-dimensional gaseous detonation velocity, one should consider a system configuration similar

266

Combustion

Velocities with wave fixed in lab space u2

u1

0

Wave direction

Burned gas

in lab flame

u1 u2

Unburned gas 0

u1

Actual laboratory velocities or velocities with respect to the tube FIGURE 5.1

Velocities used in analysis of detonation problem.

to that given in Chapter 4. For the configuration given there, it is important to understand the various velocity symbols used. Here, the appropriate velocities are defined in Fig. 5.1. With these velocities, the integrated conservation and static equations are written as ρ1u1  ρ2 u2

(5.1)

P1  ρ1u12  P2  ρ2 u22

(5.2)

c pT1  21 u12  q  c pT2  21 u22

(5.3)

P1  ρ1 RT1

(connects known variables)

(5.4)

P2  ρ2 RT2 In this type of representation, all combustion events are collapsed into a discontinuity (the wave). Thus, the unknowns are u1, u2, ρ2, T2, and P2. Since there are four equations and five unknowns, an eigenvalue cannot be obtained. Experimentally it is found that the detonation velocity is uniquely constant for a given mixture. In order to determine all unknowns, one must know something about the internal structure (rate of reaction), or one must obtain another necessary condition, which is the case for the detonation velocity determination.

1. Characterization of the Hugoniot Curve and the Uniqueness of the C–J Point The method of obtaining a unique solution, or the elimination of many of the possible solutions, will be deferred at present. In order to establish the argument for the nonexistence of various solutions, it is best to pinpoint or define the various velocities that arise in the problem and then to develop certain relationships that will prove convenient.

267

Detonation

First, one calculates expressions for the velocities, u1 and u2. From Eq. (5.1), u2  (ρ1 /ρ2 )u1 Substituting in Eq. (5.2), one has ρ1u12  (ρ12 ρ2 )u12  (P2  P1 ) 2 Dividing by ρ1 , one obtains

⎛1 1 ⎞ P P u12 ⎜⎜⎜  ⎟⎟⎟  2 2 1 , ⎜⎝ ρ1 ρ2 ⎟⎠ ρ1 1 ⎡ u12  2 ⎢⎢ (P2  P1 ) ρ1 ⎢⎣

⎞⎤ ⎛1 ⎜⎜  1 ⎟⎟⎥ ⎟ ⎜⎝⎜ ρ ρ2 ⎟⎠⎥⎥⎦ 1

(5.5)

Note that Eq. (5.5) is the equation of the Rayleigh line, which can also be derived without involving any equation of state. Since (ρ1u1)2 is always a positive value, it follows that if ρ2 ρ1, P2 P1 and vice versa. Since the sound speed c can be written as c12  γ RT1  γ P1 (1/ρ1 ) where γ is the ratio of specific heats, ⎞ ⎛P γ M12  ⎜⎜⎜ 2  1⎟⎟⎟ ⎟⎠ ⎜⎝ P1

⎡ ⎤ ⎢1  (1/ρ2 ) ⎥ ⎢ (1/ρ1 ) ⎥⎦ ⎣

(5.5a)

Substituting Eq. (5.5) into Eq. (5.1), one obtains ⎞⎤ ⎛1 ⎜⎜  1 ⎟⎟⎥ ⎜⎜⎝ ρ ρ2 ⎟⎟⎠⎥⎥⎦ 1

(5.6)

⎡⎛ ⎞⎤ P ⎞ ⎛ (1/ρ1 ) γ M22  ⎢⎢⎜⎜⎜1  1 ⎟⎟⎟ ⎜⎜⎜  1⎟⎟⎟⎥⎥ ⎟⎠⎥ P2 ⎟⎠ ⎜⎝ (1/ρ2 ) ⎢⎣⎜⎝ ⎦

(5.6a)

u22 =

1 ρ22

⎡ ⎢ (P  P ) 1 ⎢ 2 ⎢⎣

and

268

Combustion

A relationship called the Hugoniot equation, which is used throughout these developments, is developed as follows. Recall that (c p /R )  γ /(γ  1),

c p  R[γ /(γ  1)]

Substituting in Eq. (5.3), one obtains R[γ /(γ  1)]T1  21 u12  q  R[γ /(γ  1)]T2  21 u22 Implicit in writing the equation in this form is the assumption that cp and γ are constant throughout. Since RT  P/ρ, then P⎞ 1 γ ⎛⎜ P2 ⎜⎜  1 ⎟⎟⎟  (u12  u22 )  q γ  1 ⎜⎝ ρ2 ρ1 ⎟⎠ 2

(5.7)

One then obtains from Eqs. (5.5) and (5.6) ⎛ 1 ⎤ ρ2  ρ2 P2  P1 1 ⎞⎡ 1 ⎥ 2 u12  u22  ⎜⎜⎜ 2  2 ⎟⎟⎟ ⎢⎢ ⎜⎝ ρ1 ρ2 ⎟⎠ ⎣ (1/ρ1 )  (1/ρ2 ) ⎥⎦ ρ12 ρ22

⎡ ⎤ P2  P1 ⎢ ⎥ ⎢ (1/ρ )  (1/ρ ) ⎥ 1 2 ⎦ ⎣

⎞ ⎤ ⎛1 ⎛ 1 P2  P1 1 ⎞⎡ ⎥  ⎜⎜  1 ⎟⎟⎟ (P2  P1 )  ⎜⎜⎜ 2  2 ⎟⎟⎟ ⎢⎢ ⎜ ⎥ ⎜⎝ ρ1 ρ2 ⎟⎠ ⎣ (1/ρ1 )  (1/ρ2 ) ⎦ ⎜⎝ ρ1 ρ2 ⎟⎠

(5.8)

Substituting Eq. (5.8) into Eq. (5.7), one obtains the Hugoniot equation ⎛1 P⎞ 1 γ ⎛⎜ P2 1 ⎞ ⎜⎜  1 ⎟⎟⎟  (P2  P1 ) ⎜⎜⎜  ⎟⎟⎟  q ⎟ ⎜⎝ ρ1 ρ2 ⎟⎠ γ  1 ⎜⎝ ρ2 ρ1 ⎠ 2

(5.9)

This relationship, of course, will hold for a shock wave when q is set equal to zero. The Hugoniot equation is also written in terms of the enthalpy and internal energy changes. The expression with internal energies is particularly useful in the actual solution for the detonation velocity u1. If a total enthalpy (sensible plus chemical) in unit mass terms is defined such that h  c pT  h where h° is the heat of formation in the standard state and per unit mass, then a simplification of the Hugoniot equation evolves. Since by this definition q  h1  h2

269

Detonation

Eq. (5.3) becomes 1 2

u12  c pT1 + h1  c pT2  h2  21 u22 or

1 2

(u12  u22 )  h2  h1

Or further from Eq. (5.8), the Hugoniot equation can also be written as 1 2

(P2  P1 )[(1/ρ1 )  (1/ρ2 )]  h2  h1

(5.10)

To develop the Hugoniot equation in terms of the internal energy, one proceeds by first writing h  e  RT  e  ( P/ρ ) where e is the total internal energy (sensible plus chemical) per unit mass. Substituting for h in Eq. (5.10), one obtains 1 2

⎡⎛ P ⎞ ⎛ ⎞⎤ ⎢⎜⎜ 2  P2 ⎟⎟  ⎜⎜ P1  P1 ⎟⎟⎥  e  P2  e  P1 ⎟ ⎟ 2 1 ⎢⎜⎜ ρ ρ2 ρ1 ρ2 ⎟⎠ ⎜⎜⎝ ρ1 ρ2 ⎟⎠⎥⎥⎦ ⎢⎣⎝ 1 ⎛ ⎞ P P P 1 ⎜ P2 ⎜⎜  2  1  1 ⎟⎟⎟  e2  e1 ⎜ 2 ⎝ ρ1 ρ2 ρ1 ρ2 ⎟⎠

Another form of the Hugoniot equation is obtained by factoring: 1 2

(P2  P1 )[(1/ρ1 )  (1/ρ2 )]  e2  e1

(5.11)

If the energy equation [Eq. (5.3)] is written in the form h1  21 u12  h2  21 u22 the Hugoniot relations [Eqs. (5.10) and (5.11)] are derivable without the perfect gas and constant cp and γ assumptions and thus are valid for shocks and detonations in gases, metals, etc. There is also interest in the velocity of the burned gases with respect to the tube, since as the wave proceeds into the medium at rest, it is not known whether the burned gases proceed in the direction of the wave (i.e., follow the wave) or proceed away from the wave. From Fig. 5.1 it is apparent that this velocity, which is also called the particle velocity (Δu), is Δu  u1  u2

270

Combustion

and from Eqs. (5.5) and (5.6) Δu  {[(1/ρ1 )  (1/ρ2 )]( P2  P1 )}1/ 2

(5.12)

Before proceeding further, it must be established which values of the velocity of the burned gases are valid. Thus, it is now best to make a plot of the Hugoniot equation for an arbitrary q. The Hugoniot equation is essentially a plot of all the possible values of (1/ρ2, P2) for a given value of (1/ρ1, P1) and a given q. This point (1/ρ1, P1), called A, is also plotted on the graph. The regions of possible solutions are constructed by drawing the tangents to the curve that go through A [(1/ρl, P1)]. Since the form of the Hugoniot equation obtained is a hyperbola, there are two tangents to the curve through A, as shown in Fig. 5.2. The tangents and horizontal and vertical lines through the initial condition A divide the Hugoniot curve into five regions, as specified by Roman numerals (I–V). The horizontal and vertical through A are, of course, the lines of constant P and 1/ρ. A pressure difference for a final condition can be determined very readily from the Hugoniot relation [Eq. (5.9)] by considering the conditions along the vertical through A, that is, the condition of constant (1/ρ1) or constant volume heating: γ ⎛⎜ P2  P1 ⎞⎟ ⎛⎜ P2 ⎟⎜ ⎜ γ  1 ⎜⎝ ρ ⎟⎟⎠ ⎜⎝ ⎡⎛ γ ⎞ ⎤ ⎟⎟  1⎥ ⎛⎜⎜ P2 ⎢⎜⎜ ⎥ ⎜⎝ ⎢⎜⎝ γ  1 ⎟⎟⎠ ⎥⎦ ⎢⎣

 P1 ⎞⎟ ⎟ q ρ ⎟⎟⎠  P1 ⎞⎟ ⎟  q, ρ ⎟⎟⎠

(P2  P1 )  ρ(γ  1)q

(5.13)

X Z I J W II

P2

K

αJ P1

V X A 1/ρ1

III

Y

IV

1/ρ2

FIGURE 5.2 Hugoniot relationship with energy release divided into five regions (I–V) and the shock Hugoniot.

271

Detonation

From Eq. (5.13), it can be considered that the pressure differential generated is proportional to the heat release q. If there is no heat release (q  0), P1  P2 and the Hugoniot curve would pass through the initial point A. As implied before, the shock Hugoniot curve must pass through A. For different values of q, one obtains a whole family of Hugoniot curves. The Hugoniot diagram also defines an angle αJ such that tan αJ 

(P2  P1 ) (1/ρ1 )  (1/ρ2 )

From Eq. (5.5) then u1  (1/ρ1 )(tan  J )1/ 2 Any other value of αJ obtained, say by taking points along the curve from J to K and drawing a line through A, is positive and the velocity u1 is real and possible. However, from K to X, one does not obtain a real velocity due to negative αJ. Thus, region V does not represent real solutions and can be eliminated. A result in this region would require a compression wave to move in the negative direction—an impossible condition. Regions III and IV give expansion waves, which are the low-velocity waves already classified as deflagrations. That these waves are subsonic can be established from the relative order of magnitude of the numerator and denominator of Eq. (5.6a), as has already been done in Chapter 4. Regions I and II give compression waves, high velocities, and are the regions of detonation (also as established in Chapter 4). One can verify that regions I and II give compression waves and regions III and IV give expansion waves by examining the ratio of Δu to ul obtained by dividing Eq. (5.12) by the square root of Eq. (5.5): (1/ρ1 )  (1/ρ2 ) (1/ρ2 ) Δu   1 u1 (1/ρ1 ) (1/ρ1 )

(5.14)

In regions I and II, the detonation branch of the Hugoniot curve, 1/ρ2 1/ρ1 and the right-hand side of Eq. (5.14) is positive. Thus, in detonations, the hot gases follow the wave. In regions III and IV, the deflagration branch of the Hugoniot curve, 1/ρ2 1/ρ1 and the right-hand side of Eq. (5.14) is negative. Thus, in deflagrations the hot gases move away from the wave. Thus far in the development, the deflagration, and detonation branches of the Hugoniot curve have been characterized and region V has been eliminated. There are some specific characteristics of the tangency point J that were initially postulated by Chapman [7] in 1889. Chapman established that the slope of the adiabat is exactly the slope through J, that is, ⎧⎪ ⎡ ⎡ (P  P ) ⎤ ⎤ ⎫⎪ 2 1 ⎢ ⎥   ⎪⎨ ⎢ ∂P2 ⎥ ⎪⎬ ⎢ (1/ρ )  (1/ρ ) ⎥ ⎪⎪ ⎢⎣ ∂(1/ρ2 ) ⎥⎦ ⎪⎪ 1 2 ⎦J ⎣ s⎪ ⎪⎩ ⎭J

(5.15)

272

Combustion

The proof of Eq. (5.15) is a very interesting one and is verified in the following development. From thermodynamics one can write for every point along the Hugoniot curve T2 ds2  de2  P2 d (1/ρ2 )

(5.16)

where s is the entropy per unit mass. Differentiating Eq. (5.11), the Hugoniot equation in terms of e is

de2  

1 1 (P1  P2 )d (1/ρ2 )  [(1/ρ1 )  (1/ρ2 )]dP2 2 2

since the initial conditions e1, P1, and (1/ρ1) are constant. Substituting this result in Eq. (5.16), one obtains T2 ds2   21 (P1  P2 )d (1/ρ2 )  21 [(1/ρ1 )  (1/ρ2 )]dP2  P2 d (1/ρ2 )   21 (P1  P2 )d (1/ρ2 )  21 [(1/ρ1 )  (1/ρ2 )dP2

(5.17)

It follows from Eq. (5.17) that along the Hugoniot curve, ⎡ ds ⎤ ⎛ ⎞ ⎪⎧ ⎡ dP ⎤ ⎪⎫⎪ P1  P2 2 2 ⎥ ⎥  1 ⎜⎜ 1  1 ⎟⎟⎟ ⎪⎨ ⎢ T2 ⎢⎢  ⎜⎜⎝ ρ ⎥ ⎟⎠ ⎪⎪ (1/ρ1 )  (1/ρ2 ) ⎢ d (1/ρ2 ) ⎥ ⎬⎪⎪ d ( 1 / ρ ) 2 ρ 2 1 2 ⎣ ⎦H ⎣ ⎦ H ⎪⎭ ⎪⎩

(5.18)

The subscript H is used to emphasize that derivatives are along the Hugoniot curve. Now, somewhere along the Hugoniot curve, the adiabatic curve passing through the same point has the same slope as the Hugoniot curve. There, ds2 must be zero and Eq. (5.18) becomes ⎧⎪ ⎡ dP ⎤ ⎫⎪ (P1  P2 ) ⎪⎢ 2 ⎥ ⎪ ⎨⎢ ⎬  ⎥ ⎪⎪ ⎣ d (1/ρ2 ) ⎦ ⎪⎪ (1/ρ1 )  (1/ρ2 ) H⎪ ⎪⎩ ⎭s

(5.19)

But notice that the right-hand side of Eq. (5.19) is the value of the tangent that also goes through point A; therefore, the tangency point along the Hugoniot curve is J. Since the order of differentiation on the left-hand side of Eq. (5.19) can be reversed, it is obvious that Eq. (5.15) has been developed.

273

Detonation

Equation (5.15) is useful in developing another important condition at point J. The velocity of sound in the burned gas can be written as ⎛ ∂P ⎞ 1 c22  ⎜⎜⎜ 2 ⎟⎟⎟   2 ⎜⎝ ∂ρ2 ⎟⎠ ρ 2 s

⎡ ∂P ⎤ 2 ⎥ ⎢ ⎢ ∂(1/ρ ) ⎥ 2 ⎦s ⎣

(5.20)

Cross-multiplying and comparing with Eq. (5.15), one obtains ⎡ (P  P ) ⎤ ⎡ ∂P ⎤ 2 ⎥ 2 1 ⎥ ⎢ ρ22 c22   ⎢⎢  ⎢ (1/ρ )  (1/ρ ) ⎥ ⎥ ( / ) ∂ 1 ρ 2 1 2 ⎦J ⎣ ⎦s ⎣ or

[c22 ]J 

1 ρ22

⎡ (P  P ) ⎤ 2 1 2 ⎢ ⎥ ⎢ (1/ρ )  (1/ρ ) ⎥  [u2 ]J 1 2 ⎦J ⎣

Therefore [u2 ]J  [c2 ]J or [M2 ]J  1 Thus, the important result is obtained that at J the velocity of the burned gases (u2) is equal to the speed of sound in the burned gases. Furthermore, an identical analysis would show, as well, that [M2 ]Y  1 Recall that the velocity of the burned gas with respect to the tube (Δu) is written as Δu  u1  u2 or at J u1  Δu  u2 ,

u1  Δu  c2

(5.21)

Thus, at J the velocity of the unburned gases moving into the wave, that is, the detonation velocity, equals the velocity of sound in the gases behind the

274

Combustion

detonation wave plus the mass velocity of these gases (the velocity of the burned gases with respect to the tube). It will be shown presently that this solution at J is the only solution that can exist along the detonation branch of the Hugoniot curve for actual experimental conditions. Although the complete solution of u1 at J will not be attempted at this point, it can be shown readily that the detonation velocity has a simple expression now that u2 and c2 have been shown to be equal. The conservation of mass equation is rewritten to show that ρ1u1  ρ2 u2  ρ2 c2

or

ρ2 (1/ρ1 ) c  c2 ρ1 2 (1/ρ2 )

(5.21a)

1/ 2 ⎞⎟ ⎧⎪⎪ ⎡ ∂P ⎤ ⎫⎪⎪ 2 ⎥ ⎟⎟ ⎨ ⎢ ⎟ ⎪ ⎢ ∂(1/ρ2 ) ⎥ ⎬⎪⎪ 1 ⎠⎪ ⎦⎭ ⎩ ⎣

(5.22)

u1 

Then from Eq. (5.20) for c2, it follows that 1/ 2

⎪⎧⎪ ⎡ ∂P ⎤ ⎪⎫⎪ (1/ρ1 ) 2 ⎥ u1  (1/ρ2 ) ⎨ ⎢⎢ ⎬ ⎪⎪ ⎣ ∂(1/ρ2 ) ⎥⎦ ⎪⎪ (1/ρ2 ) s⎪ ⎪⎩ ⎭

⎛1  ⎜⎜⎜ ⎜⎝ ρ

With the condition that u2  c2 at J, it is possible to characterize the different branches of the Hugoniot curve in the following manner:

Region I: Region II: Region III: Region IV:

Strong detonation since P2 PJ (supersonic flow to subsonic) Weak detonation since P2 PJ (supersonic flow to supersonic) Weak deflagration since P2 PY (subsonic flow to subsonic) Strong deflagration since P2 PY (1/ρ2 1/ρY) (subsonic flow to supersonic)

At points above J, P2 PJ; thus, u2 u2,J. Since the temperature increases somewhat at higher pressures, c2 c2,J [c  (γRT)1/2]. More exactly, it is shown in the next section that above J, c2 u2. Thus, M2 above J must be less than 1. Similar arguments for points between J and K reveal M2 M2,J and hence supersonic flow behind the wave. At points past Y, 1/ρ2 1/ρ1, or the velocities are greater than u2,Y. Also past Y, the sound speed is about equal to the value at Y. Thus, past Y, M2 1. A similar argument shows that M2 1 between X and Y. Thus, past Y, the density decreases; therefore, the heat addition prescribes that there be supersonic outflow. But, in a constant area duct, it is not possible to have heat addition and proceed past the sonic condition. Thus, region IV is not a physically possible region of solutions and is ruled out.

Detonation

275

Region III (weak deflagration) encompasses the laminar flame solutions that were treated in Chapter 4. There is no condition by which one can rule out strong detonation; however, Chapman stated that in this region only velocities corresponding to J are valid. Jouguet [8] gave the following analysis. If the final values of P and 1/ρ correspond to a point on the Hugoniot curve higher than the point J, it can be shown (next section) that the velocity of sound in the burned gases is greater than the velocity of the detonation wave relative to the burned gases because, above J, c2 is shown to be greater than u2. (It can also be shown that the entropy is a minimum at J and that MJ is greater than values above J.) Consequently, if a rarefaction wave due to any reason whatsoever starts behind the wave, it will catch up with the detonation front; u1  Δu  u2. The rarefaction will then reduce the pressure and cause the final value of P2 and 1/ρ2 to drop and move down the curve toward J. Thus, points above J are not stable. Heat losses, turbulence, friction, etc., can start the rarefaction. At the point J, the velocity of the detonation wave is equal to the velocity of sound in the burned gases plus the mass velocity of these gases, so that the rarefaction will not overtake it; thus, J corresponds to a “self-sustained” detonation. The point and conditions at J are referred to as the C–J results. Thus, it appears that solutions in region I are possible, but only in the transient state, since external effects quickly break down this state. Some investigators have claimed to have measured strong detonations in the transient state. There also exist standing detonations that are strong. Overdriven detonations have been generated by pistons, and some investigators have observed oblique detonations that are overdriven. The argument used to exclude points on the Hugoniot curve below J is based on the structure of the wave. If a solution in region II were possible, there would be an equation that would give results in both region I and region II. The broken line in Fig. 5.2 representing this equation would go through A and some point, say Z, in region I and another point, say W, in region II. Both Z and W must correspond to the same detonation velocity. The same line would cross the shock Hugoniot curve at point X. As will be discussed in Section E, the structure of the detonation is a shock wave followed by chemical reaction. Thus, to detail the structure of the detonation wave on Fig. 5.2, the pressure could rise from A to X, and then be reduced along the broken line to Z as there is chemical energy release. To proceed to the weak detonation solution at W, there would have to be further energy release. However, all the energy is expended for the initial mixture at point Z. Hence, it is physically impossible to reach the solution given by W as long as the structure requires a shock wave followed by chemical energy release. Therefore, the condition of tangency at J provides the additional condition necessary to specify the detonation velocity uniquely. The physically possible solutions represented by the Hugoniot curve, thus, are only those shown in Fig. 5.3.

276

Combustion

I

J P2

III Y 1/ρ2 FIGURE 5.3 The only physical possible steady-state results along the Hugoniot—the point J and region III. The broken line represents transient conditions.

2. Determination of the Speed of Sound in the Burned Gases for Conditions above the C–J Point a. Behavior of the Entropy along the Hugoniot Curve Equation (18) was written as ⎪⎫⎪ ⎤ ⎡ d2s ⎤ ⎛ ⎞ ⎪⎧ ⎡ P1  P2 ⎥  1 ⎜⎜ 1  1 ⎟⎟⎟ ⎪⎨ ⎢ dP2 ⎥  T2 ⎢⎢ ⎬ 2 ⎥ 2 ⎜⎜⎝ ρ1 ρ2 ⎟⎠ ⎪⎪ ⎢⎣ d (1/ρ2 ) ⎥⎦ H (1/ρ1 )  (1/ρ2 ) ⎪⎪ ⎣ d (1/ρ2 ) ⎦ H ⎪⎩ ⎪⎭ with the further consequence that [ds2/d(1/ρ2)]  0 at points Y and J (the latter is the C–J point for the detonation condition). Differentiating again and taking into account the fact that [ds2 d (1/ρ2 )]  0 at point J, one obtains ⎡ d2s ⎤ (1/ρ1 )  (1/ρ2 ) ⎢ ⎥  ⎢ d (1/ρ )2 ⎥ 2T2 2 ⎣ ⎦ H at J or Y

⎡ d2P ⎤ 2 ⎥ ⎢ ⎢ d (1/ρ )2 ⎥ 2 ⎣ ⎦

(5.23)

Now [d2P2/d(1/ρ2)2] 0 everywhere, since the Hugoniot curve has its concavity directed toward the positive ordinates (see formal proof later).

277

Detonation

S2

Y

1/ρ2 1/ρ1 J

FIGURE 5.4

Variation of entropy along the Hugoniot.

S2

1/ρ1

1/ρ2

FIGURE 5.5 Entropy variation for an adiabatic shock.

Therefore, at point J, [(1/ρl)  (1/ρ2)] 0, and hence the entropy is minimum at J. At point Y, [(1/ρl)  (1/ρ2)] 0, and hence s2 goes through a maximum. When q  0, the Hugoniot curve represents an adiabatic shock. Point 1(P1, ρl) is then on the curve and Y and J are 1. Then [(1/ρl)  (1/ρ2)]  0, and the classical result of the shock theory is found; that is, the shock Hugoniot curve osculates the adiabat at the point representing the conditions before the shock. Along the detonation branch of the Hugoniot curve, the variation of the entropy is as given in Fig. 5.4. For the adiabatic shock, the entropy variation is as shown in Fig. 5.5.

b. The Concavity of the Hugoniot Curve Solving for P2 in the Hugoniot relation, one obtains P2 

a  b(1/ρ2 ) c  d (1/ρ2 )

278

Combustion

where a  q

γ  1 P1 , 2(γ  1) ρ1

b 

1 P1 , 2

c 

1 1 ρ1 , 2

d

γ 1 (5.24) 2(γ  1)

From this equation for the pressure, it is obvious that the Hugoniot curve is a hyperbola. Its asymptotes are the lines ⎛ γ  1 ⎞⎟ ⎛⎜ 1 ⎞⎟ 1 ⎟ ⎜ ⎟ 0,  ⎜⎜ ⎜⎝ γ  1 ⎟⎟⎠ ⎜⎜⎝ ρ1 ⎟⎟⎠ ρ2

P2  

γ 1 P1 0 γ 1

The slope is ⎡ dP ⎤ bc  ad 2 ⎥ ⎢ ⎢ d (1/ρ ) ⎥  [c  d (1/ρ )]2 2 ⎦H 2 ⎣ where ⎤ ⎡ γ 1 P γ ⎥ 0 bc  ad   ⎢⎢ q 1 ρ1 (γ  1)2 ⎥⎦ ⎣ 2(γ  1)

(5.25)

since q 0, P1 0, and ρl 0. A complete plot of the Hugoniot curves with its asymptotes would be as shown in Fig. 5.6. From Fig. 5.6 it is seen, as could be seen from earlier figures, that the part of the hyperbola representing the

P2

J

P1, 1/ρ1 Y (

FIGURE 5.6

γ 1 )P γ 1 1

Asymptotes to the Hugoniot curves.

γ 1

1

( γ  1 )( ρ ) 1

1/ρ2

279

Detonation

strong detonation branch has its concavity directed upward. It is also possible to determine directly the sign of ⎡ d2P ⎤ 2 ⎥ ⎢ ⎢ d (1/ρ )2 ⎥ 2 ⎣ ⎦H By differentiating Eq. (5.24), one obtains d 2 P2 2 d (ad  bc)  d (1/ρ2 )2 [c  d (1/ρ2 )3 ] Now, d 0, ad  bc 0 [Eq. (5.25)], and ⎛ 1 ⎞ 1 ⎡ γ  1 ⎛ 1 ⎞⎟ ⎛ 1 ⎞⎟⎤ ⎜⎜ ⎟  ⎜⎜ ⎟⎥ 0 c  d ⎜⎜⎜ ⎟⎟⎟  ⎢⎢ ⎜⎝ ρ2 ⎟⎠ 2 ⎢ γ  1 ⎜⎜⎝ ρ2 ⎟⎟⎠ ⎜⎜⎝ ρ1 ⎟⎟⎠⎥⎥ ⎦ ⎣ The solutions lie on the part of the hyperbola situated on the right-hand side of the asymptote (1/ρ2 )  [(γ  1)/(γ  1)](1/ρ1 ) Hence [d 2 P2 d (1/ρ2 )2 ] 0

c. The Burned Gas Speed Here ⎛ ∂s ⎞⎟ ⎛ 1 ⎞⎟ ⎛ ∂s ⎞ ⎟⎟ dP ⎟⎟ d ⎜⎜ ⎟⎟  ⎜⎜ ds  ⎜⎜⎜ ⎝ ∂(1/ρ ) ⎟⎠P ⎜⎝ ρ ⎟⎠ ⎜⎝ ∂P ⎟⎠1/ρ

(5.26)

Since ds  0 for the adiabat, Eq. (5.26) becomes ⎡ ∂s ⎤ ⎞ ⎛ ⎥  ⎜⎜ ∂s ⎟⎟ 0⎢ ⎢ ∂(1/ρ ) ⎥ ⎜⎝ ∂P ⎟⎠ 1/ρ ⎣ ⎦P

⎡ ∂P ⎤ ⎢ ⎥ ⎢ ∂(1/ρ ) ⎥ ⎣ ⎦s

(5.27)

280

Combustion

Differentiating Eq. (5.26) along the Hugoniot curve, one obtains ⎡ ds ⎤ ⎡ ⎤ ⎛ ⎞ ⎢ ⎥  ⎢ ∂s ⎥ + ⎜⎜ ∂s ⎟⎟ ⎟ ⎢ d (1/ρ ) ⎥ ⎢ ∂(1/ρ ) ⎥ ⎜ ⎣ ⎦H ⎣ ⎦ P ⎝ ∂P ⎠1/ρ

⎡ dP ⎤ ⎢ ⎥ ⎢ d (1/ρ ) ⎥ ⎣ ⎦H

(5.28)

Subtracting and transposing Eqs. (5.27) and (5.28) one has ⎡ dP ⎤ ⎡ ⎤ [ds/d (1/ρ )]H ⎢ ⎥  ⎢ ∂P ⎥  ⎢ d (1/ρ ) ⎥ ⎢ ∂(1/ρ ) ⎥ (∂s/∂P )1/ρ ⎣ ⎦H ⎣ ⎦s

(5.29)

A thermodynamic expression for the enthalpy is dh  T ds  dP/ρ

(5.30)

With the conditions of constant cp and an ideal gas, the expressions dh  c p dT ,

T  P/Rρ,

c p = [γ/(γ  1)]R

are developed and substituted in ⎛1⎞ ⎛ ∂h ⎞⎟ ∂h dh  ⎜⎜ d ⎜⎜⎜ ⎟⎟⎟ ⎟⎟ dP  ⎜⎝ ∂P ⎠ ∂(1/ρ ) ⎝ ρ ⎟⎠ to obtain ⎛ γ ⎞⎟ ⎡⎛ 1 ⎞⎟ P ⎛ 1 ⎞⎤ ⎟⎟ dP  d ⎜⎜ ⎟⎟⎟⎥⎥ ⎟⎟ R ⎢⎢⎜⎜ dh  ⎜⎜⎜ ⎜ ⎟ ⎟ R ⎜⎝ ρ ⎟⎠⎦⎥ ⎝ γ  1 ⎠ ⎢⎣⎝ Rρ ⎠ Combining Eqs. (5.30) and (5.31) gives ⎡⎛ γ ⎞ 1 ⎤ ⎟⎟ dP  dP  ⎛⎜⎜ γ ⎞⎟⎟ P d ⎛⎜⎜ 1 ⎟⎟⎞⎥ ⎢⎜⎜ ⎟ ⎟ ⎟ ⎢⎜⎝ γ  1 ⎟⎠ ρ ⎜⎝ ρ ⎟⎠⎥ ⎜⎝ γ  1 ⎟⎠ ρ ⎢⎣ ⎥⎦ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 1 1 ⎟ dP ⎜ γ ⎟ ⎟ ⎟⎟ ρ RT d ⎜⎜ ⎟⎟⎟⎥⎥  ⎢⎢⎜⎜⎜  ⎜⎜ ⎜⎝ ρ ⎟⎠ T ⎢⎣⎝ γ  1 ⎟⎟⎠ ρ ⎝ γ  1 ⎟⎠ ⎥⎦ ⎛ 1 ⎞⎟ ⎛ γ ⎞⎟ dP ⎟ Rρ d ⎜⎜ ⎟⎟   ⎜⎜ ⎜⎝ ρ ⎟⎠ (γ  1)ρT ⎜⎝ γ  1 ⎟⎟⎠

ds 

1 T

(5.31)

281

Detonation

Therefore, (∂s/∂P )1/ρ  1/(γ  1)ρT

(5.32)

Then substituting in the values of Eq. (5.32) into Eq. (5.29), one obtains ⎡ ∂P ⎤ ⎡ ⎤ ⎢ ⎥  ⎢ ∂P ⎥  (γ  1)ρT ⎢ ∂(1/ρ ) ⎥ ⎢ ⎥ ⎣ ⎦ H ⎣ ∂(1/ρ ) ⎦ s

⎡ ∂s ⎤ ⎢ ⎥ ⎢ ∂(1/ρ ) ⎥ ⎣ ⎦H

(5.33)

Equation (5.18) may be written as ⎡ ∂P ⎤ P1  P2 2T2 2 ⎥ ⎢ ⎢ ∂(1/ρ ) ⎥  (1/ρ )  (1/ρ )  (1/ρ )  (1/ρ ) 2 ⎦H 1 2 1 2 ⎣

⎡ ∂s ⎤ 2 ⎢ ⎥ ⎢ ∂(1/ρ ) ⎥ 2 ⎦H ⎣

(5.34)

Combining Eqs. (5.33) and (5.34) gives ⎡ ⎤ P2  P1 ⎢ ∂P2 ⎥  ⎢ ∂(1/ρ ) ⎥ ( 1 / ρ 2 ⎦s 1 )  (1/ρ2 ) ⎣ ⎡ ∂s ⎤ 2 ⎥  ⎢⎢ ⎥ ∂ ( 1 / ρ 2 ) ⎦H ⎣

⎡ ⎤ 2T2 ⎢ ⎥  ( γ  1 ) ρ T 2 2 ⎢ (1/ρ )  (1/ρ ) ⎥ 1 2 ⎣ ⎦

or ρ22 c22  ρ22 u22 

P2 R

⎤ ⎤⎡ ⎡ ⎢ γ  1  (ρ1 /ρ2 ) ⎥ ⎢ ∂s2 ⎥ ⎢ ⎥ ⎢ 1  (ρ1 /ρ2 ) ⎦ ⎣ ∂(1/ρ2 ) ⎥⎦ H ⎣

Since the asymptote is given by 1/ρ2 = [(γ  1)/(γ  1)](1/ρ1 ) values of (1/ρ2) on the right-hand side of the asymptote must be 1/ρ2 [(γ  1)/(γ  1)](1/ρ1 ) which leads to ⎡ ⎤ ⎢ γ  1  (ρ1 /ρ2 ) ⎥ 0 ⎢ 1  (ρ1 /ρ2 ) ⎥⎦ ⎣

(5.35)

282

Combustion

Since also [∂s/∂(1/ρ2)] 0, the right-hand side of Eq. (5.35) is the product of two negative numbers, or a positive number. If the right-hand side of Eq. (5.35) is positive, c2 must be greater than u2; that is, c2 u2

3. Calculation of the Detonation Velocity With the background provided, it is now possible to calculate the detonation velocity for an explosive mixture at given initial conditions. Equation (5.22) 1/ 2 ⎛ 1 ⎞⎡ dP2 ⎤⎥ u1  ⎜⎜⎜ ⎟⎟⎟ ⎢⎢ ⎜⎝ ρ1 ⎟⎠ ⎣ d (1/ρ2 ) ⎥⎦ s

(5.22)

shows the strong importance of density of the initial gas mixture, which is reflected more properly in the molecular weight of the products, as will be derived later. For ideal gases, the adiabatic expansion law is Pv v  constant  P2 (1/ρ2 )γ2 Differentiating this expression, one obtains ⎞⎟γ2 ⎟⎟ dP2  P2 (1/ρ2 )γ2 1γ 2 d (1/ρ2 )  0 ⎟ 2⎠

⎛ 1 ⎜⎜ ⎜⎜⎝ ρ which gives

⎡ dP ⎤ 2 ⎥  P2 γ 2  ⎢⎢ ⎥ d (1/ ρ ) (1/ρ2 ) 2 ⎦s ⎣ Substituting Eq. (5.36) into Eq. (5.22), one obtains u1 

(1/ρ1 ) (1/ρ1 ) [γ 2 P2 (1/ρ2 )]1/ 2  (γ 2 RT2 )1/ 2 (1/ρ2 ) (1/ρ2 )

If one defines μ  (1/ρ1 )/(1/ρ2 )

(5.36)

283

Detonation

then u1  μ(γ 2 RT2 )1/ 2

(5.37)

Rearranging Eq. (5.5), it is possible to write P2  P1  u12

(1/ρ1 )  (1/ρ2 ) (1/ρ1 )2

Substituting for u12 from above, one obtains (P2  P1 )

(1/ρ2 ) ⎛⎜ 1 1 ⎞  ⎜⎜  ⎟⎟⎟ ⎜⎝ ρ1 ρ2 ⎟⎠ γ 2 P2

(5.38)

Now Eq. (5.11) was e2  e1 

1 2

(P2  P1 )[(1/ρ1 )  (1/ρ2 )]

Substituting Eq. (5.38) into Eq. (5.11), one has e2  e1 =

(P  P1 )(1/ρ2 ) 1 (P2  P1 ) 2 2 γ 2 P2

or e2  e1 

1 (P22  P12 )(1/ρ2 ) 2 γ 2 P2

Since P22 P12, e2  e1 

1 P22 (1/ρ2 ) 1 P2 (1/ρ2 )  2 γ 2 P2 2 γ2

Recall that all expressions are in mass units; therefore, the gas constant R is not the universal gas constant. Indeed, it should now be written R2 to indicate this condition. Thus e2  e1 

1 P2 (1/ρ2 ) 1 R2T2  2 γ2 2 γ2

(5.39)

284

Combustion

Recall, as well, that e is the sum of the sensible internal energy plus the internal energy of formation. Equation (5.39) is the one to be solved in order to obtain T2, and hence u1. However, it is more convenient to solve this expression on a molar basis, because the available thermodynamic data and stoichiometric equations are in molar terms. Equation (5.39) may be written in terms of the universal gas constant R as e2  e1 

1 (R/MW2 )(T2 /γ 2 ) 2

(5.40)

where MW2 is the average molecular weight of the products. The gas constant R used throughout this chapter must be the engineering gas constant since all the equations developed are in terms of unit mass, not moles. R specifies the universal gas constant. If one multiplies through Eq. (5.40) with MW1, the average molecular weight of the reactants, one obtains (MW1 /MW2 )e2 (MW2 )  (MW1 )e1 

1 2

(RT2 /γ 2 )(MW1 /MW2 )

or n2 E2  E1 

1 2

(n2 RT2 /γ 2 )

(5.41)

where the E’s are the total internal energies per mole of all reactants or products and n2 is (MW1/MW2), which is the number of moles of the product per mole of reactant. Usually, one has more than one product and one reactant; thus, the E’s are the molar sums. Now to solve for T2, first assume a T2 and estimate ρ2 and MW2, which do not vary substantially for burned gas mixtures. For these approximations, it is possible to determine 1/ρ2 and P2. If Eq. (5.38) is multiplied by (P1  P2), (P1  P1 ){(1/ρ1 )  (1/ρ2 )}  (P22  P12 )(1/ρ2 )/γ 2 P2 Again P22 P12, so that P1 P P P P (1/ρ2 )  1 2  2  2 ρ1 ρ2 ρ1 ρ2 γ2 P2 P P (1/ρ2 ) P2 P  1  2   1 ρ1 ρ1 ρ2 γ2 ρ2

285

Detonation

or P2 ρ2 Pρ RT  1 1  2 2  R2T2  R1T1 ρ2 ρ1 ρ1ρ2 γ2 ⎡ (1/ρ ) ⎤ ⎡ (1/ρ ) ⎤ 1 ⎥ 2 ⎥ ⎢ R2T2 ⎢⎢ ⎥  R1T1 ⎢ (1/ρ ) ⎥  R2T2  R1T1 1 ρ ( / ) 2 1 ⎦ ⎣ ⎣ ⎦ In terms of μ, R2T2 μ  R1T1 (1/μ)  [(1/γ 2 )  1]R2T2  R1T1 which gives μ 2  [(1/γ 2 )  1  (R1T1 R2T2 )]μ  (R1T1 R2T2 )  0

(5.42)

This quadratic equation can be solved for μ; thus, for the initial condition (1/ρl), (1/ρ2) is known. P2 is then determined from the ratio of the state equations at 2 and 1: ⎛ MW T ⎞⎟ 1 2⎟ P2  μ(R2T2 R1T1 )P1 = μ ⎜⎜⎜ P ⎜⎝ MW2T1 ⎟⎟⎠ 1

(5.42a)

Thus, for the assumed T2, P2 is known. Then it is possible to determine the equilibrium composition of the burned gas mixture in the same fashion as described in Chapter 1. For this mixture and temperature, both sides of Eq. (5.39) or (5.41) are deduced. If the correct T2 was assumed, both sides of the equation will be equal. If not, reiterate the procedure until T2 is found. The correct γ2 and MW2 will be determined readily. For the correct values, u1 is determined from Eq. (5.37) written as ⎛ γ RT ⎞1/ 2 u1  μ ⎜⎜⎜ 2 2 ⎟⎟⎟ ⎜⎝ MW2 ⎟⎠

(5.42b)

The physical significance of Eq. (5.42b) is that the detonation velocity is proportional to (T2/MW2)1/2; thus it will not maximize at the stoichiometric mixture, but at one that is more likely to be fuel-rich. The solution is simpler if the assumption P2 P1 is made. Then from Eq. (5.38) ⎛ ⎞ ⎛1 ⎞ ⎜⎜  1 ⎟⎟  1 ⎜⎜ 1 ⎟⎟ , ⎟ ⎟⎟ ⎜⎜⎝ ρ ρ2 ⎠ γ 2 ⎜⎜⎝ ρ2 ⎟⎠ 1

⎛ 1 ⎜⎜ ⎜⎜⎝ ρ

⎞⎟ γ2 ⎟⎟  ⎟  γ2 1 2⎠

⎛1 ⎜⎜ ⎜⎜⎝ ρ

⎞⎟ ⎟⎟ , ⎟ 1⎠

μ

γ2  1 γ2

286

Combustion

Since one can usually make an excellent guess of γ2, one obtains μ immediately and, thus, P2. Furthermore, μ does not vary significantly for most detonation mixtures, particularly when the oxidizer is air. It is a number close to 1.8, which means, as Eq. (5.21a) specifies, that the detonation velocity is 1.8 times the sound speed in the burned gases. Gordon and McBride [12] present a more detailed computational scheme and the associated computational program.

D. COMPARISON OF DETONATION VELOCITY CALCULATIONS WITH EXPERIMENTAL RESULTS In the previous discussion of laminar and turbulent flames, the effects of the physical and chemical parameters on flame speeds were considered and the trends were compared with the experimental measurements. It is of interest here to recall that it was not possible to calculate these flame speeds explicitly; but, as stressed throughout this chapter, it is possible to calculate the detonation velocity accurately. Indeed, the accuracy of the theoretical calculations, as well as the ability to measure the detonation velocity precisely, has permitted some investigators to calculate thermodynamic properties (such as the bond strength of nitrogen and heat of sublimation of carbon) from experimental measurements of the detonation velocity. In their book, Lewis and von Elbe [13] made numerous comparisons between calculated detonation velocities and experimental values. This book is a source of such data. Unfortunately, most of the data available for comparison purposes were calculated long before the advent of digital computers. Consequently, the theoretical values do not account for all the dissociation that would normally take place. The data presented in Table 5.2 were abstracted from Lewis and von Elbe [13] and were so chosen to emphasize some important points about the factors that affect the detonation velocity. Although the agreement between the calculated and experimental values in Table 5.2 can be considered quite good, there is no doubt that the agreement would be much better if dissociation of all possible species had been allowed for in the final condition. These early data are quoted here because there have been no recent similar comparisons in which the calculated values were determined for equilibrium dissociation concentrations using modern computational techniques. Some data from Strehlow [14] are shown in Table 5.3, which provides a comparison of measured and calculated detonation velocities. The experimental data in both tables have not been corrected for the infinite tube diameter condition for which the calculations hold. This small correction would make the general agreement shown even better. Note that all experimental results are somewhat less than the calculated values. The calculated results in Table 5.3 are the more accurate ones because they were obtained by using the Gordon– McBride [12] computational program, which properly accounts for dissociation in the product composition. Shown in Table 5.4 are further calculations

287

Detonation

TABLE 5.2 Detonation Velocities of Stoichiometric Hydrogen–Oxygen Mixturesa u1 (m/s)

P2 (atm)

T2 (K)

Calculated

Experimental

(2H2  O2)

18.05

3583

2806

2819

(2H2  O2)  5 O2

14.13

2620

1732

1700

(2H2  O2)  5 N2

14.39

2685

1850

1822

(2H2  O2)  4 H2

15.97

2975

3627

3527

(2H2  O2)  5 He

16.32

3097

3617

3160

(2H2  O2)  5 Ar

16.32

3097

1762

1700

P0  1 atm, T0  291 K.

a

TABLE 5.3 Detonation Velocities of Various Mixturesa Calculated Measured velocity (m/s)

Velocity (m/s)

P2 (atm)

T2 (K)

4H2  O2

3390

3408

17.77

3439

2H2  O2

2825

2841

18.56

3679

H2  3O2

1663

1737

14.02

2667

CH4  O2

2528

2639

31.19

3332

CH4  1.5 O2

2470

2535

31.19

3725

0.7C2N2  O2

2570

2525

45.60

5210

P0  1 atm, T0  298 K.

a

of detonation parameters for propane–air and H2–air at various mixture ratios. Included in these tables are the adiabatic flame temperatures (Tad) calculated at the pressure of the burned detonation gases (P2). There are substantial differences between these values and the corresponding T2’s for the detonation condition. This difference is due to the nonisentropicity of the detonation process. The entropy change across the shock condition contributes to the additional energy term. Variations in the initial temperature and pressure should not affect the detonation velocity for a given initial density. A rise in the initial temperature could only cause a much smaller rise in the final temperature. In laminar flame

288

Combustion

TABLE 5.4 Detonation Velocities of Fuel–Air Mixtures Hydrogen–air φ  0.6

Hydrogen–air φ  1.0

Propane–air φ  0.6

1

2

1

2

1

2

M

4.44

1.00

4.84

1.00

4.64

1.00

u (m/s)

1710

973

1971

1092

1588

906

P (atm)

1.0

12.9

1.0

15.6

1.0

13.8

T (K)

298

2430

298

2947

298

2284

ρ/ρ1

1.00

1.76

1.00

1.80

1.00

1.75

Fuel–air mixture

Tad at P1 (K)

1838

2382

1701

Tad at P2 (K)

1841

2452

1702

theory, a small rise in final temperature was important since the temperature was in an exponential term. For detonation theory, recall that u1 = μ(γ 2 R2T2 )1/ 2 γ2 does not vary significantly and μ is a number close to 1.8 for many fuels and stoichiometric conditions. Examination of Table 5.2 leads one to expect that the major factor affecting u1 is the initial density. Indeed, many investigators have stated that the initial density is one of the most important parameters in determining the detonation velocity. This point is best seen by comparing the results for the mixtures in which the helium and argon inerts are added. The lower-density helium mixture gives a much higher detonation velocity than the higher-density argon mixture, but identical values of P2 and T2 are obtained. Notice as well that the addition of excess H2 gives a larger detonation velocity than the stoichiometric mixture. The temperature of the stoichiometric mixture is higher throughout. One could conclude that this variation is a result of the initial density of the mixture. The addition of excess oxygen lowers both detonation velocity and temperature. Again, it is possible to argue that excess oxygen increases the initial density. Whether the initial density is the important parameter should be questioned. The initial density appears in the parameter μ. A change in the initial density by species addition also causes a change in the final density, so that, overall, μ does not change appreciably. However, recall that R2  R/MW2

or

u1 = μ(γ 2 RT2 /MW2 )1/ 2

289

Detonation

3500

CH4/O2

3000

H2/O2 C2H2/O2

2500

u1 m/s

C3H8/O2 H2/Air

C8H18/O2

C2H2/Air

2000 CH4/Air 1500

C2H4/O2

C3H8/Air C8H18/Air

C2H4/Air

1000

500 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

φ FIGURE 5.7 The detonation velocity u1 of various fuels in air and oxygen as a function of equivalence ratio φ at initial conditions of P1  1 atm and T1  298 K.

where R is the universal gas constant and MW2 is the average molecular weight of the burned gases. It is really MW2 that is affected by initial diluents, whether the diluent is an inert or a light-weight fuel such as hydrogen. Indeed, the ratio of the detonation velocities for the excess helium and argon cases can be predicted almost exactly if one takes the square root of the inverse of the ratio of the molecular weights. If it is assumed that little dissociation takes place in these two burned gas mixtures, the reaction products in each case are two moles of water and five moles of inert. In the helium case, the average molecular weight is 9; in the argon case, the average molecular weight is 33.7. The square root of the ratio of the molecular weights is 2.05. The detonation velocity calculated for the argon mixtures is 1762. The estimated velocity for helium would be 2.05  1762  3560, which is very close to the calculated result of 3617. Since the cp’s of He and Ar are the same, T2 remains the same. The variation of the detonation velocity u1, Mach number of the detonation Ml, and the physical parameters at the C–J (burned gas) condition with equivalence ratio φ is most interesting. Figs 5.7–5.12 show this variation for hydrogen, methane, acetylene, ethene, propane, and octane detonating in oxygen and air. The data in Fig. 5.7 are interesting in that hydrogen in air or oxygen has a greater detonation velocity than any of the hydrocarbons. Indeed, there is very

290

Combustion

12

10

C8H18/O2 C3H8/O2 C2H2/O2

M1

8

C2H4/O2

CH4/O2 C8H18/Air

C2H2/Air

6 H2/O2 C3H8/Air

H2/Air 4

2 0.0

C2H4/Air

CH4/Air

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

φ FIGURE 5.8

The detonation Mach number M1 for the conditions in Fig. 5.7.

little difference in u1 between the hydrocarbons considered. The slight differences coincide with heats of formation of the fuels and hence the final temperature T2. No maximum for hydrogen was reached in Fig. 5.7 because at large φ, the molecular weight at the burned gas condition becomes very low and (T2/MW2)1/2 becomes very large. As discussed later in Section F, the rich detonation limit for H2 in oxygen occurs at φ  4.5. The rich limit for propane in oxygen occurs at φ  2.5. Since the calculations in Fig. 5.7 do not take into account the structure of the detonation wave, it is possible to obtain results beyond the limit. The same effect occurs for deflagrations in that, in a pure adiabatic condition, calculations will give values of flame speeds outside known experimental flammability limits. The order of the Mach numbers for the fuels given in Fig. 5.8 is exactly the inverse of the u1 values in Fig. 5.7. This trend is due to the sound speed in the initial mixture. Since for the calculations, T1 was always 298 K and P1 was 1 atm, the sound speed essentially varies with the inverse of the square root of the average molecular weight (MW1) of the initial mixture. For H2ßO2 mixtures, MW1 is very low compared to that of the heavier hydrocarbons. Thus the sound speed in the initial mixture is very large. What is most intriguing is to compare Figs 5.9 and 5.10. The ratio T2/T1 in the equivalence ratio range of interest does not vary significantly among the various fuels. However, there

16

14

C3H8/O2

12

C2H2/O2

H2/O2 C2H4/O2 C2H2/Air

CH4/O2

T2/T1

10

H2/Air

C8H18/O2

8

C2H4/Air

CH4/Air 6

C8H18/Air C3H8/Air

4

2 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

φ FIGURE 5.9 The ratio of the burned gas temperature T2 to the initial temperature T1 for the conditions in Fig. 5.7.

60 C8H18/O2 50 C3H8/O2 C2H2/O2

40

P2/P1

C2H4/O2 30

CH4/O2 C2H2/Air

20

H2/O2 H2/Air

C2H4/Air C8H18/Air CH4/Air C3H8/Air

10

0 0.0

0.5

1.0

1.5

2.0 φ

2.5

FIGURE 5.10 The pressure ratio P2 /P1 for the conditions in Fig. 5.7.

3.0

3.5

4.0

292

Combustion

1.90 C3H8/O2

C2H2/O2

1.85 H2/O2

C2H2/Air

CH4/O2

1.80

C2H4/O2

C8H18/O2 H2/Air

ρ2/ρ1

1.75

C2H4/Air

CH4/Air C8H18/Air

1.70

C3H8/Air 1.65

1.60

1.55 0.0 FIGURE 5.11

0.5

1.0

1.5

2.0 φ

2.5

3.0

3.5

4.0

The density ratio ρ2/ρ1 for the conditions in Fig. 5.7.

is significant change in the ratio P2/P1, particularly for the oxygen detonation condition. Indeed, there is even appreciable change in the air results; this trend is particularly important for ram accelerators operating on the detonation principle [15]. It is the P2 that applies the force in these accelerators; thus one concludes it is best to operate at near stoichiometric mixture ratios with a high-molecular-weight fuel. Equation (5.42a) explicitly specifies this P2 trend. The results of the calculated density ratio (ρ2/ρl) again reveal there is very little difference in this value for the various fuels (see Fig. 5.11). The maximum values are close to 1.8 for air and 1.85 for oxygen at the stoichiometric condition. Since μ is approximately the same for all fuels and maximizes close to φ  1, and, since the temperature ratio is nearly the same, Eq. (5.42a) indicates that it is the ratio of (MW1/MW2) that determines which fuel would likely have the greatest effect in determining P2 (see Fig. 5.12). MW2 decreases slightly with increasing φ for all fuels except H2 and is approximately the same for all hydrocarbons. MW1 decreases with φ mildly for hydrocarbons with molecular weights less than that of air or oxygen. Propane and octane, of course, increase with φ. Equation (5.42a) clearly depicts what determines P2, and indeed it appears that the average molecular weight of the unburned gas mixtures is a major factor [16]. A physical interpretation as to the molecular-weight effect can be

293

Detonation

3.5 C8H18/O2 3.0

MW1/MW2

2.5 C3H8/O2 C2H4/O2

CH4/O2

2.0 C3H8/Air

C2H2/O2

1.5

C8H18/Air

C2H4/Air C2H2/Air

1.0

H2/Air CH4/Air

H2/O2 0.5 0.0

0.5

1.0

1.5

2.0 φ

2.5

3.0

3.5

4.0

FIGURE 5.12 The average molecular weight ratio MW1/MW2 for the conditions in Fig. 5.7.

related to Ml. As stated, the larger the molecular weight of the unburned gases, the larger the M1. Considering the structure of the detonation to be discussed in the next section, the larger the P2, the larger the pressure behind the driving shock of the detonation, which is given the symbol P1 . Thus the reacting mixture that starts at a higher pressure is most likely to achieve the highest C–J pressure P2. However, for practically all hydrocarbons, particularly those in which the number of carbon atoms is greater than 2, MW2 and T2 vary insignificantly with fuel type at a given φ (see Figs. 5.9 and 5.12). Thus as a first approximation, the molecular weight of the fuel (MW1), in essence, determines P2 (Eq. 5.42a). This approximation could have ramifications in the choice of the hydrocarbon fuel for some of the various detonation type airbreathing engines being proposed. It is rather interesting that the maximum specific impulse of a rocket propellant system occurs when (T2/MW2)1/2 is maximized, even though the rocket combustion process is not one of detonation [17].

E. THE ZND STRUCTURE OF DETONATION WAVES Zeldovich [9], von Neumann [10], and Döring [11] independently arrived at a theory for the structure of the detonation wave. The ZND theory states that the

294

Combustion

detonation wave consists of a planar shock that moves at the detonation velocity and leaves heated and compressed gas behind it. After an induction period, the chemical reaction starts; and as the reaction progresses, the temperature rises and the density and pressure fall until they reach the C–J values and the reaction attains equilibrium. A rarefaction wave whose steepness depends on the distance traveled by the wave then sets in. Thus, behind the C–J shock, energy is generated by thermal reaction. When the detonation velocity was calculated in the previous section, the conservation equations were used and no knowledge of the chemical reaction rate or structure was necessary. The wave was assumed to be a discontinuity. This assumption is satisfactory because these equations placed no restriction on the distance between a shock and the seat of the generating force. But to look somewhat more closely at the structure of the wave, one must deal with the kinetics of the chemical reaction. The kinetics and mechanism of reaction give the time and spatial separation of the front and the C–J plane. The distribution of pressure, temperature, and density behind the shock depends upon the fraction of material reacted. If the reaction rate is exponentially accelerating (i.e., follows an Arrhenius law and has a relatively large overall activation energy like that normally associated with hydrocarbon oxidation), the fraction reacted changes very little initially; the pressure, density, and temperature profiles are very flat for a distance behind the shock front and then change sharply as the reaction goes to completion at a high rate. Figure 5.13, which is a graphical representation of the ZND theory, shows the variation of the important physical parameters as a function of spatial

23

13

P

P/P1 11 9 T/T1 7 ρ/ρ1

T

Shock

Induction period

5

Chemical reaction

3 ρ 1 1 1

2

1.0 cm FIGURE 5.13 Variation of physical parameters through a typical detonation wave (see Table 5.5).

295

Detonation

TABLE 5.5 Calculated Values of the Physical Parameters for Various Hydrogen– and Propane–Air/Oxygen Detonations 1

1

2

4.86 2033 1 298 1.00

0.41 377 28 1548 5.39

1.00 1129 16 2976 1.80

5.29 2920 1 298 1.00

0.40 524 33 1773 5.57

1.00 1589 19 3680 1.84

5.45 1838 1 298 1.00

0.37 271 35 1556 6.80

1.00 1028 19 2805 1.79

C3H8/O2 (φ  2.0) M u (m/s) P (atm) T (K) ρ/ρ1

8.87 2612 1 298 1.00

0.26 185 92 1932 14.15

1.00 1423 45 3548 1.84

C3H8/O2 (φ  2.2) M u (m/s) P (atm) T (K) ρ/ρ1

8.87 2603 1 298 1.00

0.26 179 92 1884 14.53

1.00 1428 45 3363 1.82

H2/Air (φ  1.2) M u (m/s) P (atm) T (K) ρ/ρ1 H2/O2 (φ  1.1) M u (m/s) P (atm) T (K) ρ/ρ1 C3H8/Air (φ  1.3) M u (m/s) P (atm) T (K) ρ/ρ1

distribution. Plane 1 is the shock front, plane 1 is the plane immediately after the shock, and plane 2 is the C–J plane. In the previous section, the conditions for plane 2 were calculated and u1 was obtained. From u1 and the shock relationships or tables, it is possible to determine the conditions at plane 1. Typical results are shown in Table 5.5 for various hydrogen and propane detonation conditions. Note from this table that (ρ2/ρl)  1.8. Therefore, for

296

Combustion

many situations the approximation that μ1 is 1.8 times the sound speed, c2, can be used. Thus, as the gas passes from the shock front to the C–J state, its pressure drops about a factor of 2, the temperature rises about a factor of 2, and the density drops by a factor of 3. It is interesting to follow the model on a Hugoniot plot, as shown in Fig. 5.14. There are two alternative paths by which a mass element passing through the wave from ε  0 to ε  1 may satisfy the conservation equations and at the same time change its pressure and density continuously, not discontinuously, with a distance of travel. The element may enter the wave in the state corresponding to the initial point and move directly to the C–J point. However, this path demands that this reaction occur everywhere along the path. Since there is little compression along this path, there cannot be sufficient temperature to initiate any reaction. Thus, there is no energy release to sustain the wave. If on another path a jump is made to the upper point (1), the pressure and temperature conditions for initiation of reaction are met. In proceeding from 1 to 1, the pressure does not follow the points along the shock Hugoniot curve. The general features of the model in which a shock, or at least a steep pressure and temperature rise, creates conditions for reaction and in which the subsequent energy release causes a drop in pressure and density have been verified by measurements in a detonation tube [18]. Most of these experiments were measurements of density variation by x-ray absorption. The possible effect of reaction rates on this structure is depicted in Fig. 5.14 as well [19]. The ZND concepts consider the structure of the wave to be one-dimensional and are adequate for determining the “static” parameters μ, ρ2, T2, and P2.

P2

1

2

Fast kinetics

Slow kinetics H curve, ε  1 ε  0.5 H curve, shock ε  0, q  0 1/ρ2 FIGURE 5.14 Effect of chemical reaction rates on detonation structures as viewed on Hugoniot curves; ε is fractional amount of chemical energy converted.

Detonation

297

However, there is now evidence that all self-sustaining detonations have a three-dimensional cellular structure.

F. THE STRUCTURE OF THE CELLULAR DETONATION FRONT AND OTHER DETONATION PHENOMENA PARAMETERS 1. The Cellular Detonation Front An excellent description of the cellular detonation front, its relation to chemical rates and their effect on the dynamic parameters, has been given by Lee [6]. With permission, from the Annual Review of Fluid Mechanics, Volume 16, © 1984 by Annual Reviews Inc., this description is reproduced almost verbatim here. Figure 5.15 shows the pattern made by the normal reflection of a detonation on a glass plate coated lightly with carbon soot, which may be from either

FIGURE 5.15 End-on pattern from the normal reflection of a cellular detonation on a smoked glass plate (after Lee [2]).

298

Combustion

0

Time (μs)

5

10

15

20

25

0

5 Distance (cm)

10

FIGURE 5.16 Laser-Schlieren chromatography of a propagating detonation in low-pressure mixtures with fish-scale pattern on a soot-covered window (courtesy of A. K. Oppenheim).

a wooden match or a kerosene lamp. The cellular structure of the detonation front is quite evident. If a similarly soot-coated polished metal (or mylar) foil is inserted into a detonation tube, the passage of the detonation wave will leave a characteristic “fish-scale” pattern on the smoked foil. Figure 5.16 is a sequence of laser-Schlieren records of a detonation wave propagating in a rectangular tube. One of the side windows has been coated with smoke, and the fish-scale pattern formed by the propagating detonation front itself is illustrated by the interferogram shown in Fig. 5.17. The direction of propagation of the detonation is toward the right. As can be seen in the sketch at the top left corner, there are two triple points. At the first triple point A, AI and AM represent the incident shock and Mach stem of the leading front, while AB is the reflected shock. Point B is the second triple point of another three-shock Mach configuration on the reflected shock AB: the entire shock pattern represents what is generally referred to as a double Mach reflection. The hatched lines denote the reaction front, while the dash–dot lines represent the shear discontinuities or slip lines associated with the triple-shock Mach configurations. The entire front ABCDE is generally referred to as the transverse wave, and it propagates normal to the direction of the detonation motion (down in the present case) at about the sound speed of the hot product gases. It has been shown conclusively that it is the triple-point regions at A and B that “write” on the smoke foil. The

299

Detonation

M

5 6

C

4 B

3 2 A 0

1 D E

l

FIGURE 5.17 Interferogram of the detailed double Mach-reflection configurations of the structure of a cellular front (courtesy of D. H. Edwards).

exact mechanics of how the triple-point region does the writing is not clear. It has been postulated that the high shear at the slip discontinuity causes the soot particles to be erased. Figure 5.17 shows a schematic of the motion of the detonation front. The fish-scale pattern is a record of the trajectories of the triple points. It is important to note the cyclic motion of the detonation front. Starting at the apex of the cell at A, the detonation shock front is highly overdriven, propagating at about 1.6 times the equilibrium C–J detonation velocity. Toward the end of the cell at D, the shock has decayed to about 0.6 times the C–J velocity before it is impulsively accelerated back to its highly overdriven state when the transverse waves collide to start the next cycle again. For the first half of the propagation from A to BC, the wave serves as the Mach stem to the incident shocks of the adjacent cells. During the second half from BC to D, the wave then becomes the incident shock to the Mach stems of the neighboring cells. Details of the variation of the shock strength and chemical reactions inside a cell can be found in a paper by Libouton et al. [20].

300

Combustion

AD is usually defined as the length Lc of the cell, and BC denotes the cell diameter (also referred to as the cell width or the transverse-wave spacing). The average velocity of the wave is close to the equilibrium C–J velocity. We thus see that the motion of a real detonation front is far from the steady and one-dimensional motion given by the ZND model. Instead, it proceeds in a cyclic manner in which the shock velocity fluctuates within a cell about the equilibrium C–J value. Chemical reactions are essentially complete within a cycle or a cell length. However, the gas dynamic flow structure is highly threedimensional; and full equilibration of the transverse shocks, so that the flow becomes essentially one-dimensional, will probably take an additional distance of the order of a few more cell lengths. From both the cellular end-on or the axial fish-scale smoke foil, the average cell size λ can be measured. The end-on record gives the cellular pattern at one precise instant. The axial record, however, permits the detonation to be observed as it travels along the length of the foil. It is much easier by far to pick out the characteristic cell size λ from the axial record; thus, the end-on pattern is not used, in general, for cell-size measurements. Early measurements of the cell size have been carried out mostly in lowpressure fuel–oxygen mixtures diluted with inert gases such as He, Ar, and N2 [21]. The purpose of these investigations is to explore the details of the detonation structure and to find out the factors that control it. It was not until very recently that Bull et al. [22] made some cell-size measurements in stoichiometric fuel–air mixtures at atmospheric pressure. Due to the fundamental importance of the cell size in the correlation with the other dynamic parameters, a systematic program has been carried out by Kynstantas to measure the cell size of atmospheric fuel–air detonations in all the common fuels (e.g., H2, C2H2, C2H4, C3H6, C2H6, C3H8, C4H10, and the welding fuel MAPP) over the entire range of fuel composition between the limits [23]. Stoichiometric mixtures of these fuels with pure oxygen, and with varying degrees of N2 dilution at atmospheric pressures, were also studied [24]. To investigate the pressure dependence, Knystautas et al. [24] have also measured the cell size in a variety of stoichiometric fuel–oxygen mixtures at initial pressures 10  p0  200 torr. The minimum cell size usually occurs at about the most detonable composition (φ  1). The cell size λ is representative of the sensitivity of the mixture. Thus, in descending order of sensitivity, we have C2H2, H2, C2H4, and the alkanes C3H8, C2H6, and C4H10. Methane (CH4), although belonging to the same alkane family, is exceptionally insensitive to detonation, with an estimated cell size λ  33 cm for stoichiometric composition as compared with the corresponding value of λ  5.35 cm for the other alkanes. That the cell size λ is proportional to the induction time of the mixture had been suggested by Shchelkin and Troshin [25] long ago. However, to compute an induction time requires that the model for the detonation structure be known, and no theory exists as yet for the real three-dimensional structure. Nevertheless, one can use the classical ZND model for the structure and compute an induction time or,

Detonation

301

equivalently, an induction-zone length l. While this is not expected to correspond to the cell size λ (or cell length Lc), it may elucidate the dependence of λ on l itself (e.g., a linear dependence λ  Al, as suggested by Shchelkin and Troshin). Westbrook [26,27] has made computations of the induction-zone length l using the ZND model, but his calculations are based on a constant volume process after the shock, rather than integration along the Rayleigh line. Very detailed kinetics of the oxidation processes are employed. By matching with one experimental point, the proportionality constant A can be obtained. The constant A differs for different gas mixtures (e.g., A  10.14 for C2H4, A  52.23 for H2); thus, the three-dimensional gas dynamic processes cannot be represented by a single constant alone over a range of fuel composition for all the mixtures. The chemical reactions in a detonation wave are strongly coupled to the details of the transient gas dynamic processes, with the end product of the coupling being manifested by a characteristic chemical length scale λ (or equivalently Lc) or time scale tc  l/C1 (where C1 denotes the sound speed in the product gases, which is approximately the velocity of the transverse waves) that describes the global rate of the chemical reactions. Since λ  0.6Lc and C1  D is the C–J detonation velocity, we have 0.5D, where τc,  Lc/D, which corresponds to the fact that the chemical reactions are essentially completed within one-cell length (or one cycle).

2. The Dynamic Detonation Parameters The extent to which a detonation will propagate from one experimental configuration into another determines the dynamic parameter called critical tube diameter. “It has been found that if a planar detonation wave propagating in a circular tube emerges suddenly into an unconfined volume containing the same mixture, the planar wave will transform into a spherical wave if the tube diameter d exceeds a certain critical value dc (i.e., d  dc). If d dc the expansion waves will decouple the reaction zone from the shock, and a spherical deflagration wave results” [6]. Rarefaction waves are generated circumferentially at the tube as the detonation leaves; then they propagate toward the tube axis, cool the shock-heated gases, and, consequently, increase the reaction induction time. This induced delay decouples the reaction zone from the shock and a deflagration persists. The tube diameter must be large enough so that a core near the tube axis is not quenched and this core can support the development of a spherical detonation wave. Some analytical and experimental estimates show that the critical tube diameter is 13 times the detonation cell size (dc  13λ) [6]. This result is extremely useful in that only laboratory tube measurements are necessary to obtain an estimate of dc. It is a value, however, that could change somewhat as more measurements are made. As in the case of deflagrations, a quenching distance exists for detonations; that is, a detonation will not propagate in a tube whose diameter is below a

302

Combustion

certain size or between infinitely large parallel plates whose separation distance is again below a certain size. This quenching diameter or distance appears to be associated with the boundary layer growth in the retainer configuration [5]. According to Williams [5], the boundary layer growth has the effect of an area expansion on the reaction zone that tends to reduce the Mach number in the burned gases, so the quenching distance arises from the competition of this effect with the heat release that increases this Mach number. For the detonation to exist, the heat release effect must exceed the expansion effect at the C–J plane; otherwise, the subsonic Mach number and the associated temperature and reaction rate will decrease further behind the shock front and the system will not be able to recover to reach the C–J state. The quenching distance is that at which the two effects are equal. This concept leads to the relation [5] δ *  (γ  1) H/8 where δ* is the boundary layer thickness at the C–J plane and H is the hydraulic diameter (4 times the ratio of the area to the perimeter of a duct which is the diameter of a circular tube or twice the height of a channel). Order-of-magnitude estimates of quenching distance may be obtained from the above expression if boundary layer theory is employed to estimate δ*; namely, δ *  l Re where Re is ρl(u1  u2)/μ and l is the length of the reaction zone; μ is evaluated at the C–J plane. Typically, Re  105 and l can be found experimentally and approximated as 6.5 times the cell size λ [28].

3. Detonation Limits As is the case with deflagrations, there exist mixture ratio limits outside of which it is not possible to propagate a detonation. Because of the quenching distance problem, one could argue that two sets of possible detonation limits can be determined. One is based on chemical-rate-thermodynamic considerations and would give the widest limits since infinite confinement distance is inherently assumed; the other follows extension of the arguments with respect to quenching distance given in the preceding paragraph. The quenching distance detonation limit comes about if the induction period or reaction zone length increases greatly as one proceeds away from the stoichiometric mixture ratio. Then the variation of δ* or l will be so great that, no matter how large the containing distance, the quenching condition will be achieved for the given mixture ratio. This mixture is the detonation limit. Belles [29] essentially established a pure chemical-kinetic-thermodynamic approach to estimating detonation limits. Questions have been raised about the approach, but the line of reasoning developed is worth considering. It is a fine example of coordinating various fundamental elements discussed to this point in order to obtain an estimate of a complex phenomenon.

303

Detonation

Belles’ prediction of the limits of detonability takes the following course. He deals with the hydrogen–oxygen case. Initially, the chemical kinetic conditions for branched-chain explosion in this system are defined in terms of the temperature, pressure, and mixture composition. The standard shock wave equations are used to express, for a given mixture, the temperature and pressure of the shocked gas before reaction is established (condition 1). The shock Mach number (M) is determined from the detonation velocity. These results are then combined with the explosion condition in terms of M and the mixture composition in order to specify the critical shock strengths for explosion. The mixtures are then examined to determine whether they can support the shock strength necessary for explosion. Some cannot, and these define the limit. The set of reactions that determine the explosion condition of the hydrogen– oxygen system is essentially k

1 OH  H 2 ⎯ ⎯⎯ → H2 O  H

k

2 H  O2 ⎯ ⎯⎯ → OH  O

k

3 O  H 2 ⎯ ⎯⎯ → OH  H

k

4 H  O2  M ⎯ ⎯⎯ → HO2  M

where M specifies the third body. (The M is used to distinguish this symbol from the symbol M used to specify the Mach number.) The steady-state solution shows that d (H 2 O) /dt  various terms/[k4 (M)  2 k2 ] Consequently the criterion for explosion is k4 (M)  2 k2

(5.43)

Using rate constants for k2 and k4 and expressing the third-body concentration (M) in terms of the temperature and pressure by means of the gas law, Belles rewrites Eq. (5.43) in the form 3.11 Te8550 / T /f x P  1

(5.44)

where fx is the effective mole fraction of the third bodies in the formation reaction for HO2. Lewis and von Elbe [13] give the following empirical relationship for fx: f x  fH2  0.35 fO2  0.43 f N2  0.20 fAr  1.47 fCO2

(5.45)

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Combustion

This expression gives a weighting for the effectiveness of other species as third bodies, as compared to H2 as a third body. Equation (5.44) is then written as a logarithmic expression (3.710 /T )  log10 (T/P )  log10 (3.11/f x )

(5.46)

This equation suggests that if a given hydrogen–oxygen mixture, which could have a characteristic value of f dependent on the mixture composition, is raised to a temperature and pressure that satisfy the equation, then the mixture will be explosive. For the detonation waves, the following relationships for the temperature and pressure can be written for the condition (1) behind the shock front. It is these conditions that initiate the deflagration state in the detonation wave: P1 P0  (1/α)[(M 2 /β )  1]

(5.47)

T1 /T0  [(M 2 /β )  1][β M 2  (1/γ )]/α 2 β M 2

(5.48)

where M is the Mach number, α  (γ  1)/(γ  1), and β  (γ  1)/2γ. Shock strengths in hydrogen mixtures are sufficiently low so that one does not have to be concerned with the real gas effects on the ratio of specific heats γ, and γ can be evaluated at the initial conditions. From Eq. (5.46) it is apparent that many combinations of pressure and temperature will satisfy the explosive condition. However, if the condition is specified that the ignition of the deflagration state must come from the shock wave, Belles argues that only one Mach number will satisfy the explosive condition. This Mach number, called the critical Mach number, is found by substituting Eqs. (5.47) and (5.48) into Eq. (5.46) to give 3.710α 2 β M 2  log10 T0 [(M 2 /β )  1][β M 2  (1/γ )]  f (T0 , P0 , α, M )  log10 (3.11 f x )

⎡ T [β M 2  (1/γ )] ⎤ ⎢ 0 ⎥ ⎢ ⎥ P0 αβ M 2 ⎣ ⎦ (5.49)

This equation is most readily solved by plotting the left-hand side as a function of M for the initial conditions. The logarithm term on the right-hand side is calculated for the initial mixture and M is found from the plot. The final criterion that establishes the detonation limits is imposed by energy considerations. The shock provides the mechanism whereby the combustion process is continuously sustained; however, the energy to drive the shock, that is, to heat up the unburned gas mixture, comes from the ultimate energy release in the combustion process. But if the enthalpy increases across

305

Detonation

the shock that corresponds to the critical Mach number is greater than the heat of combustion, an impossible situation arises. No explosive condition can be reached, and the detonation cannot propagate. Thus the criterion for the detonation of a mixture is Δhs  Δhc where Δhc is the heat of combustion per unit mass for the mixture and Δhs is the enthalpy rise across the shock for the critical Mach number (Mc). Thus hT1  hT0  Δhs

where

T1 = T0 [1  21 (γ  1)Mc2 ]

The plot of Δhc and Δhs for the hydrogen–oxygen case as given by Belles is shown in Fig. 5.18. Where the curves cross in Fig. 5.18, Δhc  Δhs, and the limits are specified. The comparisons with experimental data are very good, as is shown in Table 5.6. Questions have been raised about this approach to calculating detonation limits, and some believe that the general agreement between experiments and the theory as shown in Table 5.6 is fortuitous. One of the criticisms is that a given Mach number specifies a particular temperature and a pressure behind the shock. The expression representing the explosive condition also specifies a particular pressure and temperature. It is unlikely that there would be a direct correspondence of the two conditions from the different shock and explosion relationships. Equation (5.49) must give a unique result for the initial conditions because of the manner in which it was developed. Detonation limits have been measured for various fuel–oxidizer mixtures. These values and comparison with the deflagration (flammability) limits are

3200

Δhc

Δh (cal/gm)

2400

1600

Δhs

800

0

0

20 40 60 80 100 Percent hydrogen in mixture

FIGURE 5.18 Heat of combustion per unit mass (Δhc) and enthalpy rise across detonation shock (Δhs) as a function of hydrogen in oxygen (after Belles [29]).

306

Combustion

TABLE 5.6 Hydrogen Detonation Limits in Oxygen and Air Lean limit (vol %)

Rich limit (vol %)

System

Experimental

Calculated

Experimental

Calculated

H2ßO2

15

16.3

90

92.3

H2–Air

18.3

15.8

59.9

59.7

TABLE 5.7 Comparison of Deflagration and Detonation Limits Lean Deflagration

Rich Detonation

Deflagration

Detonation

4

15

94

90

H2–Air

4

18

74

59

COßO2

16

38

94

90

NH3ßO2

15

25

79

75

C3H8ßO2

2

3

55

37

H2ßO2

given in Table 5.7. It is interesting that the detonation limits are always narrower than the deflagration limits. But for H2 and the hydrocarbons, one should recall that, because of the product molecular weight, the detonation velocity has its maximum near the rich limit. The deflagration velocity maximum is always very near to the stoichiometric value and indeed has its minimum values at the limits. Indeed, the experimental definition of the deflagration limits would require this result.

G. DETONATIONS IN NONGASEOUS MEDIA Detonations can be generated in solid propellants and solid and liquid explosives. Such propagation through these condensed phase media make up another important aspect of the overall subject of detonation theory. The general Hugoniot relations developed are applicable, but a major difficulty exists in obtaining appropriate solutions due to the lack of good equations of state necessary due to the very high (105 atm) pressures generated. For details on this subject the reader is referred to any [30] of a number of books. Detonations will also propagate through liquid fuel droplet dispersions (sprays) in air and through solid–gas mixtures such as dust dispersions. Volatility of the liquid fuel plays an important role in characterizing the detonation developed. For low-volatility fuels, fracture and vaporization of the

307

Detonation

fuel droplets become important in the propagation mechanism, and it is not surprising that the velocities obtained are less than the theoretical maximum. Recent reviews of this subject can be found in Refs. [31] and [32]. Dust explosions and subsequent detonation generally occur when the dust particle size becomes sufficiently small that the heterogeneous surface reactions occur rapidly enough that the energy release rates will nearly support C–J conditions. The mechanism of propagation of this type of detonation is not well understood. Some reported results of detonations in dust dispersions can be found in Refs. [33] and [34].

PROBLEMS (Those with an asterisk require a numerical solution.) 1. A mixture of hydrogen, oxygen, and nitrogen, having partial pressures in the ratio 2:1:5 in the order listed, is observed to detonate and produce a detonation wave that travels at 1890 m/s when the initial temperature is 292 K and the initial pressure is 1 atm. Assuming fully relaxed conditions, calculate the peak pressure in the detonation wave and the pressure and temperature just after the passage of the wave. Prove that u2 corresponds to the C–J condition. Reasonable assumptions should be made for this problem. That is, assume that no dissociation occurs, that the pressure after the wave passes is much greater than the initial pressure, that existing gas dynamic tables designed for air can be used to analyze processes inside the wave, and that the specific heats are independent of pressure. 2. Calculate the detonation velocity in a gaseous mixture of 75% ozone (O3) and 25% oxygen (O2) initially at 298 K and 1 atm pressure. The only products after detonation are oxygen molecules and atoms. Take the ΔH f (O3 )  140 kJ/mol and all other thermochemical data from the JANAF tables in the appendixes. Report the temperature and pressure of the C–J point as well. For the mixture described in the previous problem, calculate the adiabatic (deflagration) temperature when the initial cold temperature is 298 K and the pressure is the same as that calculated for the C–J point. Compare and discuss the results for these deflagration and detonation temperatures. 3. Two mixtures (A and B) will propagate both a laminar flame and a detonation wave under the appropriate conditions:

A: B:

CH 4  i(0.21 O2  0.79 N 2 ) CH 4  i(0.21 O2  0.79 Ar )

308

Combustion

Which mixture will have the higher flame speed? Which will have the higher detonation velocity? Very brief explanations should support your answers. The stoichiometric coefficient i is the same for both mixtures. 4. What would be the most effective diluent to a detonable mixture to lower, or prevent, detonation possibility: carbon dioxide, helium, nitrogen, or argon? Order the expected effectiveness. 5.*Calculate the detonation velocity of an ethylene–air mixture at an equivalence ratio of 1 and initial conditions of 1 atm and 298 K. Repeat the calculations substituting the nitrogen in the air with equal amounts of He, Ar, and CO2. Explain the results. A chemical equilibrium analysis code, such as CEA from NASA, may be used for the analysis. 6.*Compare the effects of pressure on the detonation velocity of a stoichiometric propane–air mixture with the effect of pressure on the deflagration velocity by calculating the detonation velocity at pressures of 0.1, 1, 10, and 100 atm. Explain the similarities or differences in the trends. A chemical equilibrium analysis code, such as CEA from NASA, may be used for the analysis.

REFERENCES 1. 2. 3. 4.

5.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Friedman, R., J. Am. Rocket Soc. 23, 349 (1953). Lee, J. H., Annu. Rev. Phys. Chem. 38, 75 (1977). Urtiew, P. A., and Oppenheim, A. K., Proc. R. Soc. London, Ser. A. 295, 13 (1966). Fickett, W., and Davis, W. C., “Detonation.” University of California Press, Berkeley, 1979, Zeldovich, Y. B., and Kompaneets, A. S., “Theory of Detonation.” Academic Press, New York, 1960. Williams, F. A., “Combustion Theory,” Chap. 6., Benjamin-Cummins, Menlo Park, California, 1985, See also Linan, A., and Williams, F. A., “Fundamental Aspects of Combustion,” Oxford University Press, Oxford, England, 1994. Lee, J. H., Annu. Rev. Fluid Mech. 16, 311 (1984). Chapman, D. L., Philos. Mag. 47, 90 (1899). Jouguet, E., “Méchaniques des Explosifs.” Dorn, Paris, 1917. Zeldovich, Y. N., NACA Tech., Memo No. 1261 (1950). von Neumann, J., OSRD Rep., No. 549 (1942). Döring, W., Ann. Phys 43, 421 (1943). Gordon, S., and McBride, B. V., NASA [Spec. Publ.] SP NASA SP-273 (1971). Lewis, B., and von Elbe, G., “Combustion, Flames and Explosions of Gases,” 2nd Ed., Chap. 8. Academic Press, New York, 1961. Strehlow, R. A., “Fundamentals of Combustion.” International Textbook Co., Scranton, Pennsylvania, 1984. Li, C., Kailasanath, K., and Oran, E. S., Prog. Astronaut. Aeronaut. 453, 231 (1993). Glassman, I., and Sung, C.-J., East. States/Combust. Inst. Meet., Worcester, MA, Pap. No. 28 (1995). Glassman, I., and Sawyer, R., “The Performance of Chemical Propellants.” Technivision, Slough, England, 1971. Kistiakowsky, G. B., and Kydd, J. P., J. Chem. Phys. 25, 824 (1956). Hirschfelder, J. O., Curtiss, C. F., and Bird, R. B., “The Molecular Theory of Gases and Liquids.” Wiley, New York, 1954.

Detonation

309

20. Libouton, J. C., Dormal, M., and van Tiggelen, P. J., Prog. Astronaut. Aeronaut. 75, 358 (1981). 21. Strehlow, R. A., and Engel, C. D., AIAA J 7, 492 (1969). 22. Bull, D. C., Elsworth, J. E., Shuff, P. J., and Metcalfe, E., Combust. Flame 45, 7 (1982). 23. Knystautas, R., Guirao, C., Lee, J. H. S., and Sulmistras, A., “Int. Colloq. Dyn. Explos. React. Syst.” Poitiers, France, 1983. 24. Knystautas, R., Lee, J. H. S., and Guirao, C., Combust. Flame 48, 63 (1982). 25. Shchelkin, K. I., and Troshin, Y. K., “Gas dynamics of Combustion.” Mono Book Corp., Baltimore, Maryland, 1965. 26. Westbrook, C., Combust. Flame 46, 191 (1982). 27. Westbrook, C., and Urtiew, P., Proc. Combust. Inst. 19, 615 (1982). 28. Edwards, D. H., Jones, A. J., and Phillips, P. F., J. Phys. D9, 1331 (1976). 29. Belles, F. E., Proc. Combust. Inst. 7, 745 (1959). 30. Bowden, F. P., and Yoffee, A. D., “Limitation and Growth of Explosions in Liquids and Solids.” Cambridge University Press, Cambridge, England, 1952. 31. Dabora, E. K., in “Fuel-Air Explosions.” (J. H. S. Lee and C. M. Guirao, eds.), p. 245. University of Waterloo Press, Waterloo, Canada, 1982. 32. Sichel, M., in “Fuel-Air Explosions.” (J. H. S. Lee and C. M. Guirao, eds.), p. 265. University of Waterloo Press, Waterloo, Canada, 1982. 33. Strauss, W. A., AIAA J 6, 1753 (1968). 34. Palmer, K. N., and Tonkin, P. S., Combust. Flame 17, 159(1971).

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Chapter 6

Diffusion Flames A. INTRODUCTION Earlier chapters were concerned with flames in which the fuel and oxidizer are homogeneously mixed. Even if the fuel and oxidizer are separate entities in the initial stages of a combustion event and mixing occurs rapidly compared to the rate of combustion reactions, or if mixing occurs well ahead of the flame zone (as in a Bunsen burner), the burning process may be considered in terms of homogeneous premixed conditions. There are also systems in which the mixing rate is slow relative to the reaction rate of the fuel and oxidizer, in which case the mixing controls the burning rate. Most practical systems are mixingrate-controlled and lead to diffusion flames in which fuel and oxidizer come together in a reaction zone through molecular and turbulent diffusion. The fuel may be in the form of a gaseous fuel jet or a condensed medium (either liquid or solid), and the oxidizer may be a flowing gas stream or the quiescent atmosphere. The distinctive characteristic of a diffusion flame is that the burning (or fuel consumption) rate is determined by the rate at which the fuel and oxidizer are brought together in proper proportions for reaction. Since diffusion rates vary with pressure and the rate of overall combustion reactions varies approximately with the pressure squared, at very low pressures the flame formed will exhibit premixed combustion characteristics even though the fuel and oxidizer may be separate concentric gaseous streams. Figure 6.1 details how the flame structure varies with pressure for such a configuration where the fuel is a simple higher-order hydrocarbon [1]. Normally, the concentric fuel–oxidizer configuration is typical of diffusion flame processes.

B. GASEOUS FUEL JETS Diffusion flames have far greater practical application than premixed flames. Gaseous diffusion flames, unlike premixed flames, have no fundamental characteristic property, such as flame velocity, which can be measured readily; even initial mixture strength (the overall oxidizer-to-fuel ratio) has no practical meaning. Indeed, a mixture strength does not exist for a gaseous fuel jet issuing into a quiescent atmosphere. Certainly, no mixture strength exists for a single small fuel droplet burning in the infinite reservoir of a quiescent atmosphere. 311

312

Combustion

Radiation from carbon particles (luminous zone)

First weak continuous luminosity

Reaction zone (C2 and CH bands)

Reaction zone 30 mm

37 mm Luminous zone with intensely luminous zone

Luminous zone

Remainder of reaction zone

60 mm

Reaction zone

140 mm 1 cm

FIGURE 6.1 Structure of an acetylene–air diffusion flame at various pressures in mmHg (after Gaydon and Wolfhard [1]).

1. Appearance Only the shape of the burning laminar fuel jet depends on the mixture strength. If in a concentric configuration the volumetric flow rate of air flowing in the outer annulus is in excess of the stoichiometric amount required for the volumetric flow rate of the inner fuel jet, the flame that develops takes a closed, elongated form. A similar flame forms when a fuel jet issues into the quiescent atmosphere. Such flames are referred to as being overventilated. If in the concentric configuration the air supply in the outer annulus is reduced below an initial mixture strength corresponding to the stoichiometric required amount, a fan-shaped, underventilated flame is produced. The general shapes of the underventilated and overventilated flame are shown in Fig. 6.2 and are generally referred to as co-flow configurations. As will be shown later in this chapter, the actual heights vary with the flow conditions. The axial symmetry of the concentric configuration shown in Fig. 6.2 is not conducive to experimental analyses, particularly when some optical diagnostic tools or thermocouples are used. There are parametric variations in the r- and y-coordinates shown in Fig. 6.2. To facilitate experimental measurements on diffusion flames, the so-called Wolfhard–Parker two-dimensional gaseous fuel jet burner is used. Such a configuration is shown in Fig. 6.3, taken from Smyth et al. [2]; the screens shown in the figure are used to stabilize the flame. As can be seen in this figure, ideally there are no parametric variations along the length of the slot. Other types of gaseous diffusion flames are those in which the flow of the fuel and oxidizer oppose each other and are referred to as counterflow diffusion flames. The types most frequently used are shown in Fig. 6.4. Although

313

Diffusion Flames

rs

yF Overventilated Underventilated yF rj Air

Air Fuel

FIGURE 6.2 Appearance of gaseous fuel jet flames in a co-flow cylindrical configuration.

Stabilizing screens

Monochromator

Laser

L1

z

P L2 Flame zones

y

Air

CH4

x

Air

FIGURE 6.3 Two-dimensional Wolfhard–Parker fuel jet burner flame configuration (after Smyth et al. [2]).

these configurations are somewhat more complex to establish experimentally, they have definite advantages. The opposed jet configuration in which the fuel streams injected through a porous media such as that shown in Fig. 6.4 as Types I and II has two major advantages compared to co-flow fuel jet or Wolfhard– Parker burners. First, there is little possibility of oxidizer diffusion into the fuel side through the quench zone at the jet tube lip and, second, the flow configuration is more amenable to mathematical analysis. Although the aerodynamic configuration designated as Types III and IV produce the stagnation

314

Combustion

Air

Oxidant Matrix Flame

Fuel

Flame

Fuel

Type I

Type II

Porous cylinder

Porous sphere Fuel

Flame Flame

FIGURE 6.4

Air

Air

Type III

Type IV

Various counterflow diffusion flame experimental configurations.

point counter-flow diffusion flames as shown, it is somewhat more experimentally challenging. More interestingly, the stability of the process is more sensitive to the flow conditions. Generally, the mass flow rate through the porous media is very much smaller than the free stream flow. Usually, the fuel bleeds through the porous medium and the oxidizer is the free stream component. Of course, the reverse could be employed in such an arrangement. However, it is difficult to distinguish the separation of the flame and flow stagnation planes of the opposing flowing streams. Both, of course, are usually close to the porous body. This condition is alleviated by the approach shown by Types III and IV in Fig. 6.4. As one will note from these figures, for a hydrocarbon fuel opposing an oxidizer stream (generally air), the soot formation region that is created and the flame front can lie on either side of the flow stagnation plane. In essence, although most of the fuel is diverted by the stagnation plane, some molecules diffuse through the stagnation plane to create the flame on the oxidizer side. The contra condition for the oxidizer is shown as Type IV where the flame and soot region reside on the fuel side of the stagnation plane. Type IV depicts the fuel flow direction toward the flame front, which is the same path and occurring

315

Diffusion Flames

temperature in the axi-symmetric co-flow configuration. Type III shows that the bulk flow on the oxidizer side opposes the diffusion of the fuel molecules and thus the soot formation and particle growth are very much different from any real combustion application. In essence, the flow and temperature fields of Types I, II, and IV are similar. In fact, a liquid hydrocarbon droplet burning mimics Types I, II, and IV. As Kang et al. [3] have reported, counter-flow diffusion flames are located on the oxidizer side when hydrocarbons are the fuel. Appropriate dilution with inert gases of both the fuel and oxidizer streams, frequently used in the co-flow situation, can position the flame on the fuel side. It has been shown [4] that the criterion for the flame to be located on the fuel side is ⎡ m /Le1/ 2 ⎤ ⎢ O,o O ⎥ > i ⎢ m /Le1/ 2 ⎥ ⎢⎣ F,o F ⎥⎦ where mO,o is the free stream mass fraction of the oxidizer, mF,o is the free stream mass fraction of the fuel, i is the mass stoichiometric coefficient, and Le is the appropriate Lewis number (see Chapter 4, Section C2). The Le for oxygen is 1 and for hydrocarbons it is normally greater than 1. The profiles designating the flame height yF in Fig. 6.2 correspond to the stoichiometric adiabatic flame temperature as developed in Part 4 of this section. Indeed, the curved shape designates the stoichiometric adiabatic flame temperature isotherm. However, when one observes an overventilated co-flow flame in which a hydrocarbon constitutes the fuel jet, one observes that the color of the flame is distinctly different from that of its premixed counterpart. Whereas a premixed hydrocarbon–air flame is violet or blue green, the corresponding diffusion flame varies from bright yellow to orange. The color of the hydrocarbon diffusion flame arises from the formation of soot in the fuel part of the jet flame. The soot particles then flow through the reaction zone and reach the flame temperature. Some continue to burn through and after the flame zone and radiate. Due to the temperature that exists and the sensitivity of the eye to various wave lengths in the visible region of the electromagnetic spectrum, hydrocarbon–air diffusion flames, particularly those of co-flow structure appear to be yellow or orange. As will be discussed extensively in Chapter 8, Section E, the hydrocarbon fuel pyrolyzes within the jet and a small fraction of the fuel forms soot particles that grow in size and mass before entering the flame. The soot continues to burn as it passes through the stoichiometric isotherm position, which designates the fuel–air reaction zone and the true flame height. These high-temperature particles continue to burn and stay luminous until they are consumed in the surrounding flowing air. The end of the luminous yellow or orange image designates the soot burnout distance and not what one would call the stoichiometric flame temperature isotherm. For hydrocarbon diffusion flames, the

316

Combustion

visual distance from the jet port to the end of the “orange or yellow” flame is determined by the mixing of the burned gas containing soot with the overventilated airflow. Thus, the faster the velocity of the co-flow air stream, the shorter the distance of the particle burnout. Simply, the particle burnout is not controlled by the reaction time of oxygen and soot, but by the mixing time of the hot exhaust gases and the overventilated gas stream. Thus, hydrocarbon fuel jets burning in quiescent atmospheres appear longer than in an overventilated condition. Nonsooting diffusion flames, such as those found with H2, CO, and methanol are mildly visible and look very much like their premixed counterparts. Their true flame height can be estimated visually.

2. Structure Unlike premixed flames, which have a very narrow reaction zone, diffusion flames have a wider region over which the composition changes and chemical reactions can take place. Obviously, these changes are principally due to some interdiffusion of reactants and products. Hottel and Hawthorne [5] were the first to make detailed measurements of species distributions in a concentric laminar H2–air diffusion flame. Fig. 6.5 shows the type of results they obtained for a radial distribution at a height corresponding to a cross-section of the overventilated flame depicted in Fig. 6.2. Smyth et al. [2] made very detailed and accurate measurements of temperature and species variation across a Wolfhard–Parker burner in which methane was the fuel. Their results are shown in Figs. 6.6 and 6.7. The flame front can be assumed to exist at the point of maximum temperature, and indeed this point corresponds to that at which the maximum concentrations of major products (CO2 and H2O) exist. The same type of profiles would exist for a simple fuel jet issuing into quiescent air. The maxima arise due to diffusion of reactants in a direction normal to the flowing streams. It is most important to realize that, for the concentric configuration, molecular Concentration Nitrogen

Oxygen Products

Fuel Flame front

Center line

Flame front

FIGURE 6.5 Species variations through a gaseous fuel–air diffusion flame at a fixed height above a fuel jet tube.

317

Diffusion Flames

1.0 ∑x1

Mole fraction

N2

CH4

0.5

O2

0.0 10

H2O

5

0 5 Lateral position (mm)

10

FIGURE 6.6 Species variations throughout a Wolfhard–Parker methane–air diffusion flame (after Smyth et al. [2]).

2

1.0 Φ

Mole fraction  10

1

0

0.5 H2 CO

0.0 10

1

5

0 5 Lateral position (mm)

log (local equivalence ratio)

CO2

2 10

FIGURE 6.7 Additional species variations for the conditions of Fig. 6.6 (after Smyth et al. [2]).

318

Combustion

diffusion establishes a bulk velocity component in the normal direction. In the steady state, the flame front produces a flow outward, molecular diffusion establishes a bulk velocity component in the normal direction, and oxygen plus a little nitrogen flows inward toward the centerline. In the steady state, the total volumetric rate of products is usually greater than the sum of the other two. Thus, the bulk velocity that one would observe moves from the flame front outward. The oxygen between the outside stream and the flame front then flows in the direction opposite to the bulk flow. Between the centerline and the flame front, the bulk velocity must, of course, flow from the centerline outward. There is no sink at the centerline. In the steady state, the concentration of the products reaches a definite value at the centerline. This value is established by the diffusion rate of products inward and the amount transported outward by the bulk velocity. Since total disappearance of reactants at the flame front would indicate infinitely fast reaction rates, a more likely graphical representation of the radial distribution of reactants should be that given by the dashed lines in Fig. 6.5. To stress this point, the dashed lines are drawn to grossly exaggerate the thickness of the flame front. Even with finite reaction rates, the flame front is quite thin. The experimental results shown in Figs. 6.6 and 6.7 indicate that in diffusion flames the fuel and oxidizer diffuse toward each other at rates that are in stoichiometric proportions. Since the flame front is a sink for both the fuel and oxidizer, intuitively one would expect this most important observation. Independent of the overall mixture strength, since fuel and oxidizer diffuse together in stoichiometric proportions, the flame temperature closely approaches the adiabatic stoichiometric flame temperature. It is probably somewhat lower due to finite reaction rates, that is, approximately 90% of the adiabatic stoichiometric value [6] whether it is a hydrocarbon fuel or not. This observation establishes an interesting aspect of practical diffusion flames in that for an adiabatic situation two fundamental temperatures exist for a fuel: one that corresponds to its stoichiometric value and occurs at the flame front, and one that occurs after the products mix with the excess air to give an adiabatic temperature that corresponds to the initial mixture strength.

3. Theoretical Considerations The theory of premixed flames essentially consists of an analysis of factors such as mass diffusion, heat diffusion, and the reaction mechanisms as they affect the rate of homogeneous reactions taking place. Inasmuch as the primary mixing processes of fuel and oxidizer appear to dominate the burning processes in diffusion flames, the theories emphasize the rates of mixing (diffusion) in deriving the characteristics of such flames. It can be verified easily by experiments that in an ethylene–oxygen premixed flame, the average rate of consumption of reactants is about 4 mol/ cm3 s, whereas for the diffusion flame (by measurement of flow, flame height,

319

Diffusion Flames

ρAv j ≡ Dρ

dmA

dx mA ≡ ρA/ρ

q≡

λ d(cpT ) cp dx

ρAv  H

j

q

m

q

Δx

dx

Δx

dj Δx dx

j

ρv (cpT )

d( ρAv)

dq Δx dx

ρv (cpT ) ρv

dcpT dx

Δx

FIGURE 6.8 Balances across a differential element within a diffusion flame.

and thickness of reaction zone—a crude, but approximately correct approach), the average rate of consumption is only 6  105 mol/cm3 s. Thus, the consumption and heat release rates of premixed flames are much larger than those of pure mixing-controlled diffusion flames. The theoretical solution to the diffusion flame problem is best approached in the overall sense of a steady flowing gaseous system in which both the diffusion and chemical processes play a role. Even in the burning of liquid droplets, a fuel flow due to evaporation exists. This approach is much the same as that presented in Chapter 4, Section C2, except that the fuel and oxidizer are diffusing in opposite directions and in stoichiometric proportions relative to each other. If one selects a differential element along the x-direction of diffusion, the conservation balances for heat and mass may be obtained for the fluxes, as shown in Fig. 6.8. In Fig. 6.8, j is the mass flux as given by a representation of Fick’s law when there is bulk movement. From Fick’s law j  D(∂ρA /∂x ) As will be shown in Chapter 6, Section B2, the following form of j is exact; however, the same form can be derived if it is assumed that the total density does not vary with the distance x, as, of course, it actually does j  Dρ

∂ (ρ A /ρ ) ∂ mA  Dρ ∂x ∂x

where mA is the mass fraction of species A. In Fig. 6.8, q is the heat flux given by Fourier’s law of heat conduction; m A is the rate of decrease of mass of species A in the volumetric element (Δx · 1) (g/cm3 s), and H is the rate of chemical energy release in the volumetric element (Δx · 1)(cal/cm3 s).

320

Combustion

With the preceding definitions, for the one-dimensional problem defined in Fig. 6.8, the expression for conservation of a species A (say the oxidizer) is ∂ ρA ∂  ∂t ∂x

⎡ ∂ mA ⎢ (D ρ ) ⎢⎣ ∂x

⎤ ∂ ( ρA v ) ⎥  m A ⎥⎦ ∂x

(6.1)

where ρ is the total mass density, ρA the partial density of species A, and ν the bulk velocity in direction x. Solving this time-dependent diffusion flame problem is outside the scope of this text. Indeed, most practical combustion problems have a steady fuel mass input. Thus, for the steady problem, which implies steady mass consumption and flow rates, one may not only take ∂ρA/∂t as zero, but also use the following substitution: d ⎡ (ρ v)(ρA /ρ) ⎤⎦ d ( ρA v ) dmA  ⎣  (ρ v ) dx dx dx

(6.2)

The term (ρν) is a constant in the problem since there are no sources or sinks. With the further assumption from simple kinetic theory that Dρ is independent of temperature, and hence of x, Eq. (6.1) becomes Dρ

d 2 mA dmA  (ρ v )  m A dx 2 dx

(6.3)

Obviously, the same type of expression must hold for the other diffusing species B (say the fuel), even if its gradient is opposite to that of A so that Dρ

d 2 mB dmB  (ρ v )  m B  im A 2 dx dx

(6.4)

where m B is the rate of decrease of species B in the volumetric element (Δx · 1) and i the mass stoichiometric coefficient i

m B m A

The energy equation evolves as it did in Chapter 4, Section C2 to give 2 d (c p T ) λ d (c p T )  (ρ v )  H  im A H 2 c p dx dx

(6.5)

where H is the rate of chemical energy release per unit volume and H the heat release per unit mass of fuel consumed (in joules per gram), that is, m B H  H , im A H  H

(6.6)

321

Diffusion Flames

Since m B must be negative for heat release (exothermic reaction) to take place. Although the form of Eqs. (6.3)–(6.5) is the same as that obtained in dealing with premixed flames, an important difference lies in the boundary conditions that exist. Furthermore, in comparing Eqs. (6.3) and (6.4) with Eqs. (4.28) and (4.29), one must realize that in Chapter 4, the mass change symbol ω was always defined as a negative quantity. Multiplying Eq. (6.3) by iH, then combining it with Eq. (6.5) for the condition Le  1 or Dρ  (λ/cp), one obtains Dρ

d2 d ( c p T  mA H )  ( ρ v ) ( c p T  mA H )  0 dx 2 dx

(6.7)

This procedure is sometimes referred to as the Schvab–Zeldovich transformation. Mathematically, what has been accomplished is that the nonhomogeneous terms ( m and H ) have been eliminated and a homogeneous differential equation [Eq. (6.7)] has been obtained. The equations could have been developed for a generalized coordinate system. In a generalized coordinate system, they would have the form ⎡ ⎤ ⎛ λ ⎞⎟ ⎜ ∇ ⋅ ⎢⎢ (ρ v)(c pT )  ⎜⎜ ⎟⎟⎟ ∇(c pT ) ⎥⎥  H ⎜⎝ c p ⎟⎠ ⎢⎣ ⎥⎦

(6.8)

∇ ⋅ ⎡⎢ (ρ v)(m j )  ρ D∇m j ⎤⎥  m j ⎦ ⎣

(6.9)

These equations could be generalized even further (see Ref. 7) by simply writ where ho is the heat of formation per unit mass ing Σh oj m j instead of H, j at the base temperature of each species j. However, for notation simplicity— and because energy release is of most importance for most combustion and propulsion systems—an overall rate expression for a reaction of the type which follows will suffice F  φO → P

(6.10)

where F is the fuel, O the oxidizer, P the product, and φ the molar stoichiometric index. Then, Eqs. (6.8) and (6.9) may be written as ⎡ mj mj ∇ ⋅ ⎢⎢ (ρ v)  (ρ D)∇ MWj v j MWj v j ⎢⎣

⎤ ⎥M  ⎥ ⎥⎦

⎡ c pT c pT ∇ ⋅ ⎢⎢ (ρ v)  (ρ D)∇ HMWj v j HMWj v j ⎢⎣

⎤ ⎥M  ⎥ ⎥⎦

(6.11)

(6.12)

322

Combustion

where MW is the molecular weight, M  m j /MW j ν j ; ν j  φ for the oxidizer, and νj  1 for the fuel. Both equations have the form  ∇ ⋅ ⎡⎣ (ρ v)α  (ρ D∇α) ⎤⎦  M

(6.13)

where αT  cpT/HMWjνj and αj  mj /MWjνj. They may be expressed as L (α )  M

(6.14)

where the linear operation L(α) is defined as L (α)  ∇ ⋅ ⎡⎣ (ρ v)α  (ρ D)∇α ⎤⎦

(6.15)

The nonlinear term may be eliminated from all but one of the relationships L (α )  M . For example, L (α1 )  M

(6.16)

can be solved for α1, then the other flow variables can be determined from the linear equations for a simple coupling function Ω so that L(Ω)  0

(6.17)

where Ω  (αj  α1)  Ωj (j  1). Obviously if 1  fuel and there is a fuel– oxidizer system only, j  1 gives Ω  0 and shows the necessary redundancy.

4. The Burke–Schumann Development With the development in the previous section, it is now possible to approach the classical problem of determining the shape and height of a burning gaseous fuel jet in a coaxial stream as first described by Burke and Schumann and presented in detail in Lewis and von Elbe [8]. This description is given by the following particular assumptions: 1. At the port position, the velocities of the air and fuel are considered constant, equal, and uniform across their respective tubes. Experimentally, this condition can be obtained by varying the radii of the tubes (see Fig. 6.2). The molar fuel rate is then given by the radii ratio rj2 rs2  rj2 2. The velocity of the fuel and air up the tube in the region of the flame is the same as the velocity at the port.

323

Diffusion Flames

3. The coefficient of interdiffusion of the two gas streams is constant. Burke and Schumann [9] suggested that the effects of Assumptions 2 and 3 compensate for each other, thereby minimizing errors. Although D increases as T1.67 and velocity increases as T, this disparity should not be the main objection. The main objection should be the variation of D with T in the horizontal direction due to heat conduction from the flame. 4. Interdiffusion is entirely radial. 5. Mixing is by diffusion only, that is, there are no radial velocity components. 6. Of course, the general stoichiometric relation prevails. With these assumptions one may readily solve the coaxial jet problem. The only differential equation that one is obliged to consider is L(Ω)  0 with Ω  αF  αO where αF  mF /MWFνF and αO  mO/MWOνO. In cylindrical coordinates the generation equation becomes ⎛ ν ⎞⎟ ⎛⎜ ∂Ω ⎞⎟ ⎛ 1 ⎞⎟ ⎛ ∂ ⎞⎟ ⎛ ∂Ω ⎞⎟ ⎜⎜ ⎟ ⎜ ⎟  ⎜ ⎟ ⎜ ⎟ ⎜r ⎟ 0 ⎜⎝ D ⎟⎠ ⎜⎝ ∂y ⎟⎟⎠ ⎜⎜⎝ r ⎟⎠ ⎜⎜⎝ ∂r ⎟⎠ ⎜⎜⎝ ∂r ⎟⎠

(6.18)

The cylindrical coordinate terms in ∂/∂θ are set equal to zero because of the symmetry. The boundary conditions become Ω

mF,O

at y  0, 0 ≤ r ≤ rj

MWF ν F mO,O  MWO ν O

at y  0, rj ≤ r ≤ rs

and a ∂Ω/∂r  0 at r  rs, y 0. In order to achieve some physical insight to the coupling function Ω, one should consider it as a concentration or, more exactly, a mole fraction. At y  0 the radial concentration difference is ⎛ mO,O ⎞⎟ ⎟⎟  ⎜⎜⎜ MWF ν F ⎜⎝ MWO ν O ⎟⎠ mF,O mO,O   MWO ν O MWF ν F

Ωr rj 

mF,O

324

Combustion

This difference in Ω reveals that the oxygen acts as it were a negative fuel concentration in a given stoichiometric proportion, or vice versa. This result is, of course, a consequence of the choice of the coupling function and the assumption that the fuel and oxidizer approach each other in stoichiometric proportion. It is convenient to introduce dimensionless coordinates ξ

r , rs

η

yD vrs2

to define parameters c  rj/ rs and an initial molar mixture strength ν ν

mO,O MWF ν F mF,O MWO ν O

the reduced variable ⎛ MW ν ⎞⎟ F F ⎟ γ  Ω ⎜⎜⎜ ⎟ ⎜⎝ mF,O ⎟⎟⎠ Equation (6.18) and the boundary condition then become ∂γ ⎛⎜ 1 ⎞⎟ ⎛⎜ ∂ ⎞⎟ ⎛⎜ ∂γ ⎞⎟ ⎟  ⎜ ⎟ ⎜ ⎟ ⎜ξ ∂η ⎜⎝ ξ ⎟⎟⎠ ⎜⎝ ∂ξ ⎟⎟⎠ ⎜⎝ ∂ξ ⎟⎟⎠

γ  1 at η  0, 0 ≤ ξ < c; γ  ν at η  0, c ≤ ξ < 1

and ∂γ 0 ∂ξ

at ξ  1, η > 0

(6.19)

Equation (6.19) with these new boundary conditions has the known solution ⎧ ⎫  ⎛ 1 ⎞ ⎪⎪ J (cφ ) ⎪⎪ γ  (1  ν )c 2  ν  2(1  ν )c ∑ ⎜⎜⎜ ⎟⎟⎟ ⎪⎨ 1 n 2 ⎪⎬ ⎝ φn ⎟⎠ ⎪⎪ ⎡ J 0 (φn ) ⎤ ⎪⎪ n1 ⎜ ⎦ ⎪⎭ ⎪⎩ ⎣ × J 0 (φn ξ ) exp (φn2 η )

(6.20)

where J0 and J1 are the Bessel functions of the first kind (of order 0 and 1, respectively) and the φn represent successive roots of the equation J1(φ)  0 (with the ordering convention φn φn1, φ0  0). This equation gives the solution for Ω in the present problem. The flame shape is defined at the point where the fuel and oxidizer disappear, that is, the point that specifies the place where Ω  0. Hence, setting γ  0 provides a relation between ξ and η that defines the locus of the flame surface.

325

Diffusion Flames

The equation for the flame height is obtained by solving Eq. (6.20) for η after setting ξ  0 for the overventilated flame and ξ  1 for the underventilated flame (also γ  0). The resulting equation is still very complex. Since flame heights are large enough to cause the factor exp(φn2 η ) to decrease rapidly as n increases at these values of η, it usually suffices to retain only the first few terms of the sum in the basic equation for this calculation. Neglecting all terms except n  1, one obtains the rough approximation ⎛ 1 ⎞ ⎧⎪ 2(1  ν )cJ (cφ ) ⎫⎪⎪ 1 1 η  ⎜⎜⎜ 2 ⎟⎟⎟ ln ⎪⎨ ⎬ ⎜⎝ φ1 ⎟⎠ ⎪⎪ ⎡⎣ ν  (1  ν )c ⎤⎦ φ1 J 0 (φ1 ) ⎪⎪ ⎩ ⎭

(6.21)

for the dimensionless height of the overventilated flame. The first zero of J1(φ) is φ1  3.83. The flame shapes and heights predicted by such expressions (Fig. 6.9) are shown by Lewis and von Elbe [8] to be in good agreement with experimental results—surprisingly so, considering the basic drastic assumptions. Indeed, it should be noted that Eq. (6.21) specifies that the dimensionless flame height can be written as η  f (c, ν )

yF D  f (c, ν ) vrs2

(6.22)

Thus since c  (rj/rs), the flame height can also be represented by yF 

vrj2 Dc 2

f (c, ν ) 

πrj2 v πD

f (c, ν ) 

Q f (c, ν ) πD

(6.23)

1.2 1.0

0.5

0.8

0.4

0.6

0.3

0.4

0.2

0.2

0.1

0 0

0 0.2 0.4 0.6 0.8 1.0 Radial distance, x (in.)

Inches underventilated, y

Inches overventilated, y

1.4

FIGURE 6.9 Flame shapes as predicted by Burke–Schumann theory for cylindrical fuel jet systems (after Burke and Schumann [9]).

326

Combustion

where v is the average fuel velocity, Q the volumetric flow rate of the fuel, and f   (f /c2). Thus, one observes that the flame height of a circular fuel jet is directly proportional to the volumetric flow rate of the fuel. In a pioneering paper [10], Roper vastly improved on the Burke–Schumann approach to determine flame heights not only for circular ports, but also for square ports and slot burners. Roper’s work is significant because he used the fact that the Burke–Schumann approach neglects buoyancy and assumes the mass velocity should everywhere be constant and parallel to the flame axis to satisfy continuity [7], and then pointed out that resulting errors cancel for the flame height of a circular port burner but not for the other geometries [10, 11]. In the Burke–Schumann analysis, the major assumption is that the velocities are everywhere constant and parallel to the flame axis. Considering the case in which buoyancy forces increase the mass velocity after the fuel leaves the burner port, Roper showed that continuity dictates decreasing streamline spacing as the mass velocity increases. Consequently, all volume elements move closer to the flame axis, the widths of the concentration profiles are reduced, and the diffusion rates are increased. Roper [10] also showed that the velocity of the fuel gases is increased due to heating and that the gases leaving the burner port at temperature T0 rapidly attain a constant value Tf in the flame regions controlling diffusion; thus the diffusivity in the same region is D  D0 (Tf /T0 )1.67 where D0 is the ambient or initial value of the diffusivity. Then, considering the effect of temperature on the velocity, Roper developed the following relationship for the flame height: ⎛ T ⎞⎟0.67 1 Q ⎜⎜ 0 ⎟ yF  ⎟ 4π D0 ln[1  (1/S )] ⎜⎜⎝ Tf ⎟⎠

(6.24)

where S is the stoichiometric volume rate of air to volume rate of fuel. Although Roper’s analysis does not permit calculation of the flame shape, it does produce for the flame height a much simpler expression than Eq. (6.21). If due to buoyancy the fuel gases attain a velocity vb in the flame zone after leaving the port exit, continuity requires that the effective radial diffusion distance be some value rb. Obviously, continuity requires that ρ be the same for both cases, so that rj2 v  rb2 vb Thus, one observes that regardless of whether the fuel jet is momentum- or buoyancy-controlled, the flame height yF is directly proportional to the volumetric flow leaving the port exit.

327

Diffusion Flames

Given the condition that buoyancy can play a significant role, the fuel gases start with an axial velocity and continue with a mean upward acceleration g due to buoyancy. The velocity of the fuel gases v is then given by v  (v02  vb2 )1 / 2

(6.25)

where v is the actual velocity, v0 the momentum-driven velocity at the port, and vb the velocity due to buoyancy. However, the buoyancy term can be closely approximated by vb2  2 gyF

(6.26)

where g is the acceleration due to buoyancy. If one substitutes Eq. (6.26) into Eq. (6.25) and expands the result in terms of a binomial expression, one obtains ⎡ ⎤ ⎛ gy ⎞⎟ 1 ⎛ gy ⎞⎟2 ⎢ ⎥ v  v0 ⎢1  ⎜⎜⎜ 2F ⎟⎟  ⎜⎜⎜ 2F ⎟⎟  ⎥ ⎟ ⎟ ⎜⎝ v0 ⎠ 2 ⎜⎝ v0 ⎠ ⎢ ⎥ ⎢⎣ ⎥⎦

(6.27)

where the term in parentheses is the inverse of the modified Froude number ⎛ v2 ⎞ Fr ≡ ⎜⎜⎜ 0 ⎟⎟⎟ ⎜⎝ gyF ⎟⎠

(6.28)

Thus, Eq. (6.27) can be written as ⎡ ⎤ ⎛ 1 ⎞ ⎛ 1 ⎞⎟ ⎥ v  v0 ⎢1  ⎜⎜ ⎟⎟⎟  ⎜⎜ 

⎟ ⎟ ⎢ ⎥ ⎜⎝ Fr ⎠ ⎜⎝ 2 Fr 2 ⎠ ⎣ ⎦

(6.29)

For large Froude numbers, the diffusion flame height is momentum-controlled and v  v0. However, most laminar burning fuel jets will have very small Froude numbers and v  vb, that is, most laminar fuel jets are buoyancy-controlled. Although the flame height is proportional to the fuel volumetric flow rate whether the flame is momentum- or buoyancy-controlled, the time to the flame tip does depend on what the controlling force is. The characteristic time for diffusion (tD) must be equal to the time (ts) for a fluid element to travel from the port to the flame tip, that is, if the flame is momentum-controlled, ⎛ rj2 ⎞⎟ ⎛ y ⎞ ⎜ t D  ⎜⎜ ⎟⎟⎟  ⎜⎜ F ⎟⎟⎟  ts ⎜⎝ D ⎟⎠ ⎜⎝ v ⎟⎠

(6.30)

328

Combustion

It follows from Eq. (6.30), of course, that yF 

rj2 v D



πrj2 v πD



Q πD

(6.31)

Equation (6.31) shows the same dependence on Q as that developed from the Burke–Schumann approach [Eqs. (6.21)–(6.23)]. For a momentumcontrolled fuel jet flame, the diffusion distance is rj, the jet port radius; and from Eq. (6.30) it is obvious that the time to the flame tip is independent of the fuel volumetric flow rate. For a buoyancy-controlled flame, ts remains proportional to (yF / v); however, since v  (2gyF)1/2, t

yF yF   y1F/ 2  Q1/ 2 v ( yF )1/ 2

(6.32)

Thus, the stay time of a fuel element in a buoyancy-controlled laminar diffusion flame is proportional to the square root of the fuel volumetric flow rate. This conclusion is significant with respect to the soot smoke height tests to be discussed in Chapter 8. For low Froude number, a fuel leaving a circular jet port immediately becomes affected by a buoyancy condition and the flame shape is determined by this condition and the consumption of the fuel. In essence, following Eq. (6.26) the velocity within the exit fuel jet becomes proportional to its height above the port exit as the flow becomes buoyantly controlled. For example, the velocity along the axis would vary according to the proportionality v  y1/2. Thus, the velocity increases as it   ρvA, where m approaches the flame tip. Conservation of mass requires m  is the mass flow rate, ρ the density of the flowing gaseous fuel (or the fuel plus additives). Thus the flame shape contracts not only due to the consumption of the fuel, but also as the buoyant velocity increases and a conical flame shape develops just above the port exit. For a constant mass flow buoyant system v  yF1/2  QF1/2. Thus v  P1/2 and, since the mass flow must be constant, not affected by pressure change, m   ρvA  PP1/2P1/2  constant. This proportionality states that, as the pressure of an experiment is raised, the planar cross-sectional area across any part of a diffusion flame must contract and take a narrower conical shape. This pressure effect has been clearly shown experimentally by Flowers and Bowman [12]. The preceding analyses hold only for circular fuel jets. Roper [10] has shown, and the experimental evidence verifies [11], that the flame height for a slot burner is not the same for momentum- and buoyancy-controlled jets. Consider a slot burner of the Wolfhard–Parker type in which the slot width is x and the length is L. As discussed earlier for a buoyancy-controlled situation, the diffusive distance would not be x, but some smaller width, say xb. Following the terminology of Eq. (6.25), for a momentum-controlled slot burner, t

y x2  F D v0

(6.33)

329

Diffusion Flames

and v0 x 2 /D ⎛⎜ Q ⎞⎟ ⎛⎜ xb ⎞⎟  ⎜ ⎟⎟ ⎜ ⎟⎟ ⎜⎝ D ⎠ ⎜⎝ L ⎟⎠ L/L

yF 

(6.34)

For a buoyancy-controlled slot burner, t

yF 

(x b )2 y  F D vb

vb (xb )2 /D ⎛⎜ Q ⎞⎟ ⎛⎜ xb ⎞⎟  ⎜ ⎟⎟ ⎜ ⎟⎟ ⎜⎝ D ⎠ ⎜⎝ L ⎟⎠ L/L

(6.35)

(6.36)

Recalling that xb must be a function of the buoyancy, one has xb Lvb  Q x

Q Q  vb L (2 gyF )1/ 2 L

Thus Eq. (6.36) becomes ⎛ ⎞⎟2 / 3 ⎛ Q 4 ⎞⎟1/ 3 Q2 ⎟⎟  ⎜⎜ ⎟ yF  ⎜⎜⎜ 2 ⎜⎝ D 2 L2 2 g ⎟⎟⎠ ⎝ DL (2 g )1/ 2 ⎟⎠

(6.37)

Comparing Eqs. (6.34) and (6.37), one notes that under momentum-controlled conditions for a given Q, the flame height is directly proportional to the slot width while that under buoyancy-controlled conditions for a given Q, the flame height is independent of the slot width. Roper et al. [11] have verified these conclusions experimentally.

5. Turbulent Fuel Jets The previous section considered the burning of a laminar fuel jet, and the essential result with respect to flame height was that ⎛ rj2 v ⎞⎟ ⎛ Q ⎞ ⎜ yF,L  ⎜⎜ ⎟⎟  ⎜⎜ ⎟⎟ ⎜⎝ D ⎟⎟⎠ ⎜⎝ D ⎟⎠

(6.38)

where yF,L specifies the flame height. When the fuel jet velocity is increased to the extent that the flow becomes turbulent, the Froude number becomes large and the system becomes momentum-controlled—moreover, molecular diffusion considerations lose their validity under turbulent fuel jet conditions. One would intuitively expect a change in the character of the flame and its height. Turbulent

330

Combustion

flows are affected by the occurrence of flames, as discussed for premixed flame conditions, and they are affected by diffusion flames by many of the same mechanisms discussed for premixed flames. However, the diffusion flame in a turbulent mixing layer between the fuel and oxidizer streams steepens the maximum gradient of the mean velocity profile somewhat and generates vorticity of opposite signs on opposite sides of the high-temperature, low-density reaction region. The decreased overall density of the mixing layer with combustion increases the dimensions of the large vortices and reduces the rate of entrainment of fluids into the mixing layer [13]. Thus it is appropriate to modify the simple phenomenological approach that led to Eq. (6.31) to account for turbulent diffusion by replacing the molecular diffusivity with a turbulent eddy diffusivity . Consequently, the turbulent form of Eq. (6.38) becomes ⎛ rj2 v ⎞⎟ ⎜ ⎟⎟ yF,T  ⎜⎜ ⎜⎝ ε ⎟⎟⎠ where yF,T is the flame height of a turbulent fuel jet. But ε  lU , where l is the integral scale of turbulence, which is proportional to the tube diameter (or radius rj), and U is the intensity of turbulence, which is proportional to the mean flow velocity v along the axis. Thus one may assume that ε  rj v

(6.39)

Combining Eq. (6.39) and the preceding equation, one obtains yF,T 

rj2 v rj v

 rj

(6.40)

This expression reveals that the height of a turbulent diffusion flame is proportional to the port radius (or diameter) above, irrespective of the volumetric fuel flow rate or fuel velocity issuing from the burner! This important practical conclusion has been verified by many investigators. The earliest verification of yF,T  rj was by Hawthorne et al. [14], who reported their results as yF,T as a function of jet exit velocity from a fixed tube exit radius. Thus varying the exit velocity is the same as varying the Reynolds number. The results in Ref. 14 were represented by Linan and Williams [13] in the diagram duplicated here as Fig. 6.10. This figure clearly shows that as the velocity increases in the laminar range, the height increases linearly, in accordance with Eq. (6.38). After transition to turbulence, the height becomes independent of the velocity, in agreement with Eq. (6.40). Observing Fig. 6.10, one notes that the transition to turbulence begins near the top of the flame, then as the velocity increases, the turbulence rapidly recedes to the exit of the jet. At a high-enough velocity, the flow in the fuel tube becomes turbulent and turbulence is observed everywhere in the flame. Depending on the fuel mixture, liftoff usually occurs after the flame becomes fully turbulent. As the velocity increases further after liftoff, the liftoff

331

Diffusion Flames

40 Turbulent

Height (cm)

Laminar

Blowoff

Liftoff 0

0

40 Exit velocity (m/s)

80

FIGURE 6.10 Variation of the character (height) of a gaseous diffusion flame as a function of fuel jet velocity showing experimental flame liftoff (after Linan and Williams [13] and Hawthorne et al. [14]).

height (the axial distance between the fuel jet exit and the point where combustion begins) increases approximately linearly with the jet velocity [13]. After the liftoff height increases to such an extent that it reaches a value comparable to the flame diameter, a further increase in the velocity causes blowoff [13].

C. BURNING OF CONDENSED PHASES When most liquids or solids are projected into an atmosphere so that a combustible mixture is formed, an ignition of this mixture produces a flame that surrounds the liquid or solid phase. Except at the very lowest of pressures, around 106 Torr, this flame is a diffusion flame. If the condensed phase is a liquid fuel and the gaseous oxidizer is oxygen, the fuel evaporates from the liquid surface and diffuses to the flame front as the oxygen moves from the surroundings to the burning front. This picture of condensed phase burning is most readily applied to droplet burning, but can also be applied to any liquid surface. The rate at which the droplet evaporates and burns is generally considered to be determined by the rate of heat transfer from the flame front to the fuel surface. Here, as in the case of gaseous diffusion flames, chemical processes are assumed to occur so rapidly that the burning rates are determined solely by mass and heat transfer rates. Many of the early analytical models of this burning process considered a double-film model for the combustion of the liquid fuel. One film separated the droplet surface from the flame front and the other separated the flame front from the surrounding oxidizer atmosphere, as depicted in Fig. 6.11. In some analytical developments the liquid surface was assumed to be at the normal boiling point of the fuel. Surveys of the temperature field in burning liquids by Khudyakov [15] indicated that the temperature is just a few degrees below the boiling point. In the approach to be employed here, the only requirement is that the droplet be at a uniform temperature at or below the

332

Combustion

1.5

T

rA

Nitrogen

Fuel

mA  r

0.5 T T Tf

Products

0

s

f Droplet flame radius

Oxidizer

∞

FIGURE 6.11 Characteristic parametric variations of dimensionless temperature T’ and mass fraction m of fuel, oxygen, and products along a radius of a droplet diffusion flame in a quiescent atmosphere. Tf is the adiabatic, stoichiometric flame temperature, ρA is the partial density of species A, and ρ is the total mass density. The estimated values derived for benzene are given in Section 2b.

normal boiling point. In the sf region of Fig. 6.11, fuel evaporates at the drop surface and diffuses toward the flame front where it is consumed. Heat is conducted from the flame front to the liquid and vaporizes the fuel. Many analyses assume that the fuel is heated to the flame temperature before it chemically reacts and that the fuel does not react until it reaches the flame front. This latter assumption implies that the flame front is a mathematically thin surface where the fuel and oxidizer meet in stoichiometric proportions. Some early investigators first determined Tf in order to calculate the fuel-burning rate. However, in order to determine a Tf, the infinitely thin reaction zone at the stoichiometric position must be assumed. In the film f, oxygen diffuses to the flame front, and combustion products and heat are transported to the surrounding atmosphere. The position of the boundary designated by  is determined by convection. A stagnant atmosphere places the boundary at an infinite distance from the fuel surface. Although most analyses assume no radiant energy transfer, as will be shown subsequently, the addition of radiation poses no mathematical difficulty in the solution to the mass burning rate problem.

1. General Mass Burning Considerations and the Evaporation Coefficient Three parameters are generally evaluated: the mass burning rate (evaporation), the flame position above the fuel surface, and the flame temperature. The most important parameter is the mass burning rate, for it permits the evaluation of the so-called evaporation coefficient, which is most readily measured experimentally. The use of the term evaporation coefficient comes about from mass and heat transfer experiments without combustion—that is, evaporation, as generally

333

Diffusion Flames

used in spray-drying and humidification. Basically, the evaporation coefficient β is defined by the following expression, which has been verified experimentally: d 2  d02  β t

(6.41)

where d0 is the original drop diameter and d the drop diameter after time t. It will be shown later that the same expression must hold for mass and heat transfer with chemical reaction (combustion). The combustion of droplets is one aspect of a much broader problem, which involves the gasification of a condensed phase, that is, a liquid or a solid. In this sense, the field of diffusion flames is rightly broken down into gases and condensed phases. Here the concern is with the burning of droplets, but the concepts to be used are just as applicable to other practical experimental properties such as the evaporation of liquids, sublimation of solids, hybrid burning rates, ablation heat transfer, solid propellant burning, transpiration cooling, and the like. In all categories, the interest is the mass consumption rate, or the rate of regression of a condensed phase material. In gaseous diffusion flames, there was no specific property to measure and the flame height was evaluated; but in condensed phase diffusion flames, a specific quantity is measurable. This quantity is some representation of the mass consumption rate of the condensed phase. The similarity to the case just mentioned arises from the fact that the condensed phase must be gasified; consequently, there must be an energy input into the condensed material. What determines the rate of regression or evolution of material is the heat flux at the surface. Thus, in all the processes mentioned, q  rρf Lv

(6.42)

where q is the heat flux to the surface in calories per square centimeter per second, r the regression rate in centimeters per second, ρf the density of the condensed phase, and Lv the overall energy per unit mass required to gasify the material. Usually, Lv is the sum of two terms—the heat of vaporization, sublimation, or gasification plus the enthalpy required to bring the surface to the temperature of vaporization, sublimation, or gasification. From the foregoing discussion, it is seen that the heat flux q, Lv, and the density determine the regression rate; but this statement does not mean that the heat flux is the controlling or rate-determining step in each case. In fact, it is generally not the controlling step. The controlling step and the heat flux are always interrelated, however. Regardless of the process of concern (assuming no radiation), ⎛ ∂T ⎞⎟ ⎟ q  λ ⎜⎜ ⎜⎝ ∂y ⎟⎟⎠ s

(6.43)

334

Combustion

where λ is the thermal conductivity and the subscript “s” designates the fuel surface. This simple statement of the Fourier heat conduction law is of such great significance that its importance cannot be overstated. This same equation holds whether or not there is mass evolution from the surface and whether or not convective effects prevail in the gaseous stream. Even for convective atmospheres in which one is interested in the heat transfer to a surface (without mass addition of any kind, i.e., the heat transfer situation generally encountered), one writes the heat transfer equation as q  h(T  Ts )

(6.44)

Obviously, this statement is shorthand for ⎛ ∂T ⎞⎟ ⎟⎟  h(T  Ts ) q  λ ⎜⎜⎜ ⎝ ∂y ⎟⎠s

(6.45)

where T and Ts are the free-stream and surface temperatures, respectively; the heat transfer coefficient h is by definition h

λ δ

(6.46)

where δ is the boundary layer thickness. Again by definition, the boundary layer is the distance between the surface and free-stream condition; thus, as an approximation, q

λ(T  Ts ) δ

(6.47)

The (T  Ts)/δ term is the temperature gradient, which correlates (∂T/∂y)s through the boundary layer thickness. The fact that δ can be correlated with the Reynolds number and that the Colburn analogy can be applied leads to the correlation of the form Nu  f (Re, Pr )

(6.48)

where Nu is the Nusselt number (hx/λ), Pr the Prandtl number (cpμ/λ), and Re the Reynolds number (ρvx/μ); here x is the critical dimension—the distance from the leading edge of a flat plate or the diameter of a tube. Although the correlations given by Eq. (6.48) are useful for practical evaluation of heat transfer to a wall, one must not lose sight of the fact that the temperature gradient at the wall actually determines the heat flux there. In transpiration cooling problems, it is not so much that the injection of the transpiring fluid increases the boundary layer thickness, thereby decreasing the

335

Diffusion Flames

heat flux, but rather that the temperature gradient at the surface is decreased by the heat-absorbing capability of the injected fluid. What Eq. (6.43) specifies is that regardless of the processes taking place, the temperature profile at the surface determines the regression rate—whether it be evaporation, solid propellant burning, etc. Thus, all the mathematical approaches used in this type of problem simply seek to evaluate the temperature gradient at the surface. The temperature gradient at the surface is different for the various processes discussed. Thus, the temperature profile from the surface to the source of energy for evaporation will differ from that for the burning of a liquid fuel, which releases energy when oxidized in a flame structure. Nevertheless, a diffusion mechanism generally prevails; and because it is the slowest step, it determines the regression rate. In evaporation, the mechanism is the conduction of heat from the surrounding atmosphere to the surface; in ablation, it is the conduction of heat through the boundary layer; in droplet burning, it is the rates at which the fuel diffuses to approach the oxidizer, etc. It is mathematically interesting that the gradient at the surface will always be a boundary condition to the mathematical statement of the problem. Thus, the mathematical solution is necessary simply to evaluate the boundary condition. Furthermore, it should be emphasized that the absence of radiation has been assumed. Incorporating radiation transfer is not difficult if one assumes that the radiant intensity of the emitters is known and that no absorption occurs between the emitters and the vaporizing surfaces, that is, it can be assumed that qr, the radiant heat flux to the surface, is known. Then Eq. (6.43) becomes ⎛ ∂T ⎞⎟ ⎟⎟  qr  r ρf Lv q  qr  λ ⎜⎜⎜ ⎝ ∂y ⎟⎠s

(6.49)

Note that using these assumptions does not make the mathematical solution of the problem significantly more difficult, for again, qr—and hence radiation transfer—is a known constant and enters only in the boundary condition. The differential equations describing the processes are not altered. First, the evaporation rate of a single fuel droplet is calculated before considering the combustion of this fuel droplet; or, to say it more exactly, one calculates the evaporation of a fuel droplet in the absence of combustion. Since the concern is with diffusional processes, it is best to start by reconsidering the laws that hold for diffusional processes. Fick’s law states that if a gradient in concentration of species A exists, say (dnA/dy), a flow or flux of A, say jA, across a unit area in the y direction will be proportional to the gradient so that jA  D

dnA dy

(6.50)

336

Combustion

where D is the proportionality constant called the molecular diffusion coefficient or, more simply, the diffusion coefficient; nA is the number concentration of molecules per cubic centimeter; and j is the flux of molecules, in number of molecules per square centimeter per second. Thus, the units of D are square centimeters per second. The Fourier law of heat conduction relates the flux of heat q per unit area, as a result of a temperature gradient, such that q  λ

dT dy

The units of q are calories per square centimeter per second and those of the thermal conductivity λ are calories per centimeter per second per degree Kelvin. It is not the temperature, an intensive thermodynamic property, that is exchanged, but energy content, an extensive property. In this case, the energy density and the exchange reaction, which show similarity, are written as q 

λ dT λ d (ρ c p T ) dH ρc p   α ρc p dy ρc p dy dy

(6.51)

where α is the thermal diffusivity whose units are square centimeters per second since λ  cal/cm s K, cp  cal/g K, and ρ  g/cm3; and H is the energy concentration in calories per cubic centimeter. Thus, the similarity of Fick’s and Fourier’s laws is apparent. The former is due to a number concentration gradient, and the latter to an energy concentration gradient. A law similar to these two diffusional processes is Newton’s law of viscosity, which relates the flux (or shear stress) τyx of the x component of momentum due to a gradient in ux; this law is written as τ yx  μ

dux dy

(6.52)

where the units of the stress τ are dynes per square centimeter and those of the viscosity are grams per centimeter per second. Again, it is not velocity that is exchanged, but momentum; thus when the exchange of momentum density is written, similarity is again noted. ⎡ d (ρux ) ⎤ ⎛ μ ⎞ ⎡ d (ρux ) ⎤ ⎥  ν ⎢ ⎥ τ yx   ⎜⎜⎜ ⎟⎟⎟ ⎢ ⎢ ⎥ ⎢ dy ⎥ ⎟ ⎝ ρ ⎠ ⎣ dy ⎦ ⎣ ⎦

(6.53)

where ν is the momentum diffusion coefficient or, more acceptably, the kinematic viscosity; ν is a diffusivity and its units are also square centimeters per second. Since Eq. (6.53) relates the momentum gradient to a flux, its similarity

337

Diffusion Flames

to Eqs. (6.50) and (6.51) is obvious. Recall, as stated in Chapter 4, that the simple kinetic theory for gases predicts α  D  ν. The ratios of these three diffusivities give some familiar dimensionless similarity parameters. Pr 

ν ν α , Sc  , Le  α D D

where Pr, Sc, and Le are the Prandtl, Schmidt, and Lewis numbers, respectively. Thus, for gases simple kinetic theory gives as a first approximation Pr  Sc  Le  1

2. Single Fuel Droplets in Quiescent Atmospheres Since Fick’s law will be used in many different forms in the ensuing development, it is best to develop those forms so that the later developments need not be interrupted. Consider the diffusion of molecules A into an atmosphere of B molecules, that is, a binary system. For a concentration gradient in A molecules alone, future developments can be simplified readily if Fick’s law is now written jA  DAB

dnA dy

(6.54)

where DAB is the binary diffusion coefficient. Here, nA is considered in number of moles per unit volume, since one could always multiply Eq. (6.50) through by Avogadro’s number, and jA is expressed in number of moles as well. Multiplying through Eq. (6.54) by MWA, the molecular weight of A, one obtains ( jA MWA )  DAB

d (nA MWA ) dρ = DAB A dy dy

(6.55)

where ρ A is the mass density of A, ρB is the mass density of B, and n is the total number of moles. n  nA  nB

(6.56)

which is constant and dn dn dn  0, A   B , dy dy dy

jA   jB

(6.57)

338

Combustion

The result is a net flux of mass by diffusion equal to ρv  jA MWA  jB MWB

(6.58)

 jA (MWA  MWB )

(6.59)

where v is the bulk direction velocity established by diffusion. In problems dealing with the combustion of condensed matter, and hence regressing surfaces, there is always a bulk velocity movement in the gases. Thus, species are diffusing while the bulk gases are moving at a finite velocity. The diffusion of the species can be against or with the bulk flow (velocity). For mathematical convenience, it is best to decompose such flows into a flow with an average mass velocity v and a diffusive velocity relative to v. When one gas diffuses into another, as A into B, even without the quasisteady-flow component imposed by the burning, the mass transport of a species, say A, is made up of two components—the normal diffusion component and the component related to the bulk movement established by the diffusion process. This mass transport flow has a velocity ΔA and the mass of A transported per unit area is ρAΔA. The bulk velocity established by the diffusive flow is given by Eq. (6.58). The fraction of that flow is Eq. (6.58) multiplied by the mass fraction of A, ρA/ρ. Thus, ⎡ ⎛ MW ⎞⎟⎤ ⎛ ρ ⎞⎟⎫⎪⎪ ⎛ d ρ ⎞ ⎧⎪⎪ B ⎟⎥ ⎜ A ⎟ ρA ΔA  DAB ⎜⎜⎜ A ⎟⎟⎟ ⎨1  ⎢⎢1  ⎜⎜⎜ ⎟⎟⎥ ⎜⎜ ⎟⎟⎬⎪ ⎜ MW ⎝ dy ⎟⎠ ⎪⎪⎪ ⎝ A ⎠⎥⎦ ⎝ ρ ⎠⎪ ⎢⎣ ⎪⎭ ⎩

(6.60)

Since jA  jB ⎛ MW ⎞⎟ A ⎟ MWA jA  MWA jB  (MWB jB ) ⎜⎜⎜ ⎜⎝ MWB ⎟⎟⎠

(6.61)

and 

⎛ MW ⎞⎟ ⎛ d ρ ⎞ d ρA A ⎟⎜  ⎜⎜⎜ ⎜ B ⎟⎟ ⎜⎝ MWB ⎟⎟⎠ ⎜⎝ dy ⎟⎟⎠ dy

(6.62)

However, d ρA dρ dρ  B  dy dy dy

(6.63)

⎛ d ρA ⎞⎟ ⎛⎜ MWB ⎞⎟ ⎛ d ρA ⎞⎟ d ρ ⎜⎜ ⎟ ⎜⎜ ⎟ ⎟⎜ ⎜⎝ dy ⎟⎟⎠ ⎜⎜⎝ MWA ⎟⎟⎠ ⎜⎝ dy ⎟⎟⎠ dy

(6.64)

which gives with Eq. (6.62)

339

Diffusion Flames

Multiplying through by ρA/r, one obtains ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ρA ⎞⎟ ⎡ ⎜⎜ ⎟ ⎢1  MWB ⎥ ⎜⎜ d ρA ⎟⎟  ⎜⎜ ρA ⎟⎟ ⎜⎜ d ρ ⎟⎟ ⎟ ⎜⎝ ρ ⎟⎠ ⎢ MWA ⎥⎦ ⎜⎝ dy ⎟⎟⎠ ⎜⎝ ρ ⎟⎟⎠ ⎜⎝ dy ⎟⎟⎠ ⎣

(6.65)

Substituting Eq. (6.65) into Eq. (6.60), one finds ⎡ ⎛ d ρ ⎞ ⎛ ρ ⎞ ⎛ d ρ ⎞⎤ ρA ΔA  DAB ⎢⎢⎜⎜⎜ A ⎟⎟⎟  ⎜⎜⎜ A ⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟⎥⎥ ⎢⎣⎝ dy ⎟⎠ ⎝ ρ ⎟⎠ ⎝ dy ⎟⎠⎥⎦

(6.66)

or ρA ΔA  ρ DAB

d ( ρA / ρ ) dy

(6.67)

Defining mA as the mass fraction of A, one obtains the following proper form for the diffusion of species A in terms of mass fraction: ρA ΔA  ρ DAB

dmA dy

(6.68)

This form is that most commonly used in the conservation equation. The total mass flux of A under the condition of the burning of a condensed phase, which imposes a bulk velocity developed from the mass burned, is then ρA vA  ρA v  ρA ΔA  ρA v  ρ DAB

dmA dy

(6.69)

where ρAv is the bulk transport part and ρAΔA is the diffusive transport part. Indeed, in the developments of Chapter 4, the correct diffusion term was used without the proof just completed.

a. Heat and Mass Transfer without Chemical Reaction (Evaporation): The Transfer Number B Following Blackshear’s [16] adaptation of Spalding’s approach [17,18], consideration will now be given to the calculation of the evaporation of a single fuel droplet in a nonconvective atmosphere at a given temperature and pressure. A major assumption is now made in that the problem is considered as a quasi-steady one. Thus, the droplet is of fixed size and retains that size by a steady flux of fuel. One can consider the regression as being constant; or, even better, one can think of the droplet as a porous sphere being fed from a very thin tube at a rate equal to the mass evaporation rate so that the surface

340

Combustion

Δr

T dT r T dr

r

r Δr

FIGURE 6.12 Temperature balance across a differential element of a diffusion flame in spherical symmetry.

of the sphere is always wet and any liquid evaporated is immediately replaced. The porous sphere approach shows that for the diffusion flame, a bulk gaseous velocity outward must exist; and although this velocity in the spherical geometry will vary radially, it must always be the value given by m  4πr 2 ρ v. This velocity is the one referred to in the last section. With this physical picture one may take the temperature throughout the droplet as constant and equal to the surface temperature as a consequence of the quasi-steady assumption. In developing the problem, a differential volume in the vapor above the liquid droplet is chosen, as shown in Fig. 6.12. The surface area of a sphere is 4πr2. Since mass cannot accumulate in the element, d (ρ Av)  0,

(d/dr )(4πr 2 ρ v)  0

(6.70)

which is essentially the continuity equation. Consider now the energy equation of the evaporating droplet in spherical–symmetric coordinates in which cp and λ are taken independent of temperature. The heat entering at the surface  pT or (4πr 2ρv)cpT (see Fig. 6.12). (i.e., the amount of heat convected in) is mc  The heat leaving after r  Δr is mc p [T  (dT/dr )Δr ] or (4πr 2ρv)cp[T  (dT/dr)Δr]. The difference, then is 4πr 2 ρ vc p (dT/dr )Δr The heat diffusing from r toward the drop (out of the element) is λ 4πr 2 (dT/dr )

341

Diffusion Flames

The heat diffusing into the element is λ 4π(r  Δr )2 (d/dr )[T  (dT/dr )Δr ] or ⎡ ⎛ d 2T ⎞ ⎛ dT ⎞⎟⎤ ⎛ dT ⎞⎟  ⎢⎢ λ 4πr 2 ⎜⎜ ⎟⎥ ⎟⎟  λ 4πr 2 ⎜⎜⎜ 2 ⎟⎟⎟ Δr  λ8πr Δr ⎜⎜ ⎜⎝ dr ⎟⎠⎥⎥ ⎜⎝ dr ⎠ ⎝ dr ⎟⎠ ⎢⎣ ⎦ plus two terms in Δr2 and one in Δr3 which are negligible. The difference in the two terms is [λ 4πr 2 (d 2T/dr 2 )Δr  8λπr (dT/dr )Δr ] Heat could be generated in the volume element defined by Δr so one has (4πr 2 Δr ) H where H is the rate of enthalpy change per unit volume. Thus, for the energy balance 4πr 2 ρ vc p (dT/dr )Δr  λ 4πr 2 (d 2T/dr 2 )Δr  λ8πr Δr (dT/dr ) (6.71)  4πr 2 ΔrH ⎛ d 2T ⎞ ⎛ dT ⎞⎟ ⎛ dT ⎞⎟ 4πr 2 ρ vc p ⎜⎜ ⎟  4πr 2 H ⎟⎟  λ 4πr 2 ⎜⎜⎜ 2 ⎟⎟⎟  8λπr ⎜⎜ ⎜⎝ dr ⎟⎠ ⎜⎝ dr ⎠ ⎝ dr ⎟⎠

(6.72)

⎡⎛ ⎛ dc pT ⎞⎟ 2 ⎞ ⎛ dc T ⎞⎤ ⎟⎟  d ⎢⎢⎜⎜⎜ λ 4πr ⎟⎟⎟ ⎜⎜ p ⎟⎟⎟⎥⎥  4πr 2 H 4πr 2 (ρ v) ⎜⎜⎜ ⎜⎝ dr ⎟⎠ dr ⎢⎜⎝ c p ⎟⎟⎠ ⎜⎜⎝ dr ⎟⎠⎥ ⎣ ⎦

(6.73)

or

Similarly, the conservation of the Ath species can be written as ⎛ dm ⎞⎤ ⎛ dm ⎞ ⎛ d ⎞ ⎡ 4πr 2 ρ v ⎜⎜ A ⎟⎟⎟  ⎜⎜ ⎟⎟⎟ ⎢⎢ 4πr 2 ρ D ⎜⎜ A ⎟⎟⎟⎥⎥  4πr 2 m A ⎜⎝ dr ⎟⎠ ⎜⎝ dr ⎟⎠ ⎜⎝ dr ⎠ ⎣ ⎦

(6.74)

where m A is the generation or disappearance rate of A due to reaction in the unit volume. According to the kinetic theory of gases, to a first approximation the product Dρ (and hence λ/cp) is independent of temperature and pressure; consequently, Dsρs  Dρ, where the subscript s designates the condition at the droplet surface.

342

Combustion

Consider a droplet of radius r. If the droplet is vaporizing, the fluid will leave the surface by convection and diffusion. Since at the liquid droplet surface only A exists, the boundary condition at the surface is ρl r  ρs vs  Amount of material leaving the surface

⎛ dm ⎞  ρ mAs vs  ρ D ⎜⎜ A ⎟⎟⎟ ⎜⎝ dr ⎟⎠ s

(6.75)

where ρl is the liquid droplet density and r the rate of change of the liquid droplet radius. Equation (6.75) is, of course, explicitly Eq. (6.69) when ρsvs is the bulk mass movement, which at the surface is exactly the amount of A that is being convected (evaporated) written in terms of a gaseous density and velocity plus the amount of gaseous A that diffuses to or from the surface. Since products and inerts diffuse to the surface, mAs has a value less than 1. Equation (6.75), then, is written as vs 

D(dmA /dr )s (mAs  1)

(6.76)

In the sense of Spalding, a new parameter b is defined: b

mA mAs  1

Equation (6.76) thus becomes ⎛ db ⎞ vs  D ⎜⎜ ⎟⎟⎟ ⎜⎝ dr ⎠ s

(6.77)

and Eq. (6.74) written in terms of the new variable b and for the evaporation condition (i.e., m A  0) is ⎛ db ⎞ ⎛ d ⎞ ⎡ ⎛ db ⎞⎤ r 2 ρ v ⎜⎜ ⎟⎟⎟  ⎜⎜ ⎟⎟⎟ ⎢ r 2 ρ D ⎜⎜ ⎟⎟⎟⎥ ⎜⎝ dr ⎠ ⎜⎝ dr ⎠ ⎢ ⎜⎝ dr ⎠⎥ ⎣ ⎦

(6.78)

The boundary condition at r   is mA  mA or b  b

at r → 

(6.79)

From continuity r 2 ρ v  rs2 ρs vs

(6.80)

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Diffusion Flames

Since r2ρv  constant on the left-hand side of the equation, integration of Eq. (6.78) proceeds simply and yields ⎛ db ⎞ r 2 ρ vb  r 2 ρ D ⎜⎜ ⎟⎟⎟  const ⎜⎝ dr ⎠

(6.81)

Evaluating the constant at r  rs, one obtains rs2 ρs vs bs  rs2 ρs vs  const since from Eq. (6.77) vs  D(db/dr)s. Or, one has from Eq. (6.81) ⎛ db ⎞ rs2 ρs vs (b  bs  1)  r 2 ρ D ⎜⎜ ⎟⎟⎟ ⎜⎝ dr ⎠

(6.82)

By separating variables, ⎛ rs2 ρs vs ⎞⎟ db ⎜⎜ ⎟ ⎜⎜⎝ r 2 ρ D ⎟⎟⎠ dr  b  b  1 s

(6.83)

assuming ρD constant, and integrating (recall that ρD  ρsDs), one has ⎛ r2v ⎞  ⎜⎜⎜ s s ⎟⎟⎟  ln(b  bs  1) + const ⎜⎝ rDs ⎟⎠

(6.84)

Evaluating the constant at r → , one obtains const   ln(b  bs  1) or Eq. (6.84) becomes ⎡ b  b 1⎤ rs2 vs s ⎥  ln ⎢⎢  ⎥ rDs b  b s 1 ⎦ ⎣

(6.85)

The left-hand term of Eq. (6.84) goes to zero as r → . This point is significant because it shows that the quiescent spherical–symmetric case is the only mathematical case that does not blow up. No other quiescent case, such as that

344

Combustion

for cylindrical symmetry or any other symmetry, is tractable. Evaluating Eq. (6.85) at r  rs results in ⎛ r v ⎞⎟ ⎜⎜ s s ⎟  ln(b  b  1)  ln(1  B) s  ⎜⎜ D ⎟⎟ ⎝ s ⎠ rs vs  Ds ln(b  bs  1)  Ds ln(1  B) ⎡ m  m ⎤ ⎪⎫⎪ ⎪⎧ As ⎥ rs vs  Ds ln ⎪⎨1  ⎢⎢ A ⎥ ⎬⎪ ⎪⎪ m  1 As ⎣ ⎦ ⎪⎭ ⎩

(6.86)

Here B  b  bs and is generally referred to as the Spalding transfer number. The mass consumption rate per unit area GA  m A / 4πrs2, where m A  ρs vs 4πrs2, is then found by multiplying Eq. (6.86) by 4πrs2 ρs /4πrs2 and cross-multiplying rs, to give 4πrs2 ρs vs Dρ  s s ln(1  B) 4πrs2 rs ⎛ Dρ ⎞⎟ ⎛ρ D ⎞ m ⎟ ln(1  B) GA  A2  ⎜⎜⎜ s s ⎟⎟⎟ ln(1  B)  ⎜⎜⎜ ⎜⎝ rs ⎟⎟⎠ ⎜⎝ rs ⎟⎠ 4πrs

(6.87)

Since the product Dρ is independent of pressure, the evaporation rate is essentially independent of pressure. There is a mild effect of pressure on the transfer number, as will be discussed in more detail when the droplet burning case is considered. In order to find a solution for Eq. (6.87) or, more rightly, to evaluate the transfer number B, mAs must be determined. A reasonable assumption would be that the gas surrounding the droplet surface is saturated at the surface temperature Ts. Since vapor pressure data are available, the problem then is to determine Ts. For the case of evaporation, Eq. (6.73) becomes rs2 ρs vs c p

⎛ d ⎞⎡ ⎛ dT ⎞⎟⎤ dT  ⎜⎜ ⎟⎟⎟ ⎢ r 2λ ⎜⎜ ⎟⎥ ⎜⎝ dr ⎠ ⎢ ⎜⎝ dr ⎟⎠⎥ dr ⎣ ⎦

(6.88)

which, upon integration, becomes ⎛ dT ⎞⎟ rs2 ρs vs c pT  r 2λ ⎜⎜ ⎟  const ⎜⎝ dr ⎟⎠

(6.89)

The boundary condition at the surface is ⎡ ⎛ dT ⎞⎤ ⎟⎥  ρ v L ⎢ λ ⎜⎜ s s v ⎢ ⎜⎝ dr ⎟⎟⎠⎥ ⎣ ⎦s

(6.90)

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Diffusion Flames

where Lv is the latent heat of vaporization at the temperature Ts. Recall that the droplet is considered uniform throughout at the temperature Ts. Substituting Eq. (6.90) into Eq. (6.89), one obtains const  rs2 ρs vs (c pTs  Lv ) Thus, Eq. (6.89) becomes ⎡ ⎛ L ⎞⎟⎤ ⎛ dT ⎞⎟ ⎜ rs2 ρs vs c p ⎢⎢ T  Ts  ⎜⎜ v ⎟⎟⎟⎥⎥  r 2λ ⎜⎜ ⎟ ⎜⎝ dr ⎟⎠ ⎜ ⎢⎣ ⎝ c p ⎟⎠⎥⎦

(6.91)

Integrating Eq. (6.91), one has rs2 ρs vs c p rλ

⎡ ⎛ L ⎞⎟⎤ ⎜  ln ⎢⎢ T  Ts  ⎜⎜ v ⎟⎟⎟⎥⎥  const ⎜⎝ c p ⎟⎠⎥ ⎢⎣ ⎦

(6.92)

After evaluating the constant at r → , one obtains rs2 ρs vs c p rλ

⎡ T  T  ( L /c ) ⎤  s v p ⎥  ln ⎢⎢ ⎥ T − T  ( L ⎢⎣ s v /c p ) ⎥⎦

(6.93)

Evaluating Eq. (6.93) at the surface (r  rs; T  Ts) gives rs ρs vs c p λ

⎛ c p (T  Ts ) ⎞⎟  ln ⎜⎜⎜  1⎟⎟ ⎟⎠ ⎜⎝ Lv

(6.94)

And since α  λ/cpρ, ⎛ c p (T  Ts ) ⎞⎟ ⎟⎟ rs vs  αs ln ⎜⎜⎜1  ⎟⎠ ⎜⎝ Lv

(6.95)

Comparing Eqs. (6.86) and (6.95), one can write ⎡ ⎞⎟ ⎛ c p (T  Ts ) ⎛ m  m ⎞⎟⎤ As ⎟⎥ rs vs  αs ln ⎜⎜⎜  1⎟⎟  Ds ln ⎢⎢1  ⎜⎜⎜ A ⎟⎟⎥ ⎟⎠ ⎜ ⎜⎝ Lv m  1 ⎝ ⎠⎥⎦ As ⎢⎣ or rs vs  αs ln(1  BT )  Ds ln(1  BM )

(6.96)

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Combustion

where BT 

c p (T  Ts ) Lv

BM 

,

mA  mAs mAs  1

Again, since α  D, BT  BM and c p (T  Ts ) Lv



mA  mAs mAs  1

(6.97)

Although mAs is determined from the vapor pressure of A or the fuel, one must realize that mAs is a function of the total pressure since mAs 

⎛ P ⎞ ⎛ MWA ⎞⎟ ρAs n MWA  A  ⎜⎜ A ⎟⎟⎟ ⎜⎜ ⎟ ⎜⎝ P ⎟⎠ ⎜⎝ MW ⎟⎟⎠ nMW ρ

(6.98)

where nA and n are the number of moles of A and the total number of moles, respectively; MWA and MW the molecular weight of A and the average molecular weight of the mixture, respectively; and PA and P the vapor pressure of A and the total pressure, respectively. In order to obtain the solution desired, a value of Ts is assumed, the vapor pressure of A is determined from tables, and mAs is calculated from Eq. (6.98). This value of mAs and the assumed value of Ts are inserted in Eq. (6.97). If this equation is satisfied, the correct Ts is chosen. If not, one must reiterate. When the correct value of Ts and mAs are found, BT or BM are determined for the given initial conditions T or mA. For fuel combustion problems, mA is usually zero; however, for evaporation, say of water, there is humidity in the atmosphere and this humidity must be represented as mA. Once BT and BM are determined, the mass evaporation rate is determined from Eq. (6.87) for a fixed droplet size. It is, of course, much preferable to know the evaporation coefficient β from which the total evaporation time can be determined. Once B is known, the evaporation coefficient can be determined readily, as will be shown later.

b. Heat and Mass Transfer with Chemical Reaction (Droplet Burning Rates) The previous developments also can be used to determine the burning rate, or evaporation coefficient, of a single droplet of fuel burning in a quiescent atmosphere. In this case, the mass fraction of the fuel, which is always considered to be the condensed phase, will be designated mf, and the mass fraction of the oxidizer mo. mo is the oxidant mass fraction exclusive of inerts and i is

347

Diffusion Flames

used as the mass stoichiometric fuel–oxidant ratio, also exclusive of inerts. The same assumptions that hold for evaporation from a porous sphere hold here. Recall that the temperature throughout the droplet is considered to be uniform and equal to the surface temperature Ts. This assumption is sometimes referred to as one of infinite condensed phase thermal conductivity. For liquid fuels, this temperature is generally near, but slightly less than, the saturation temperature for the prevailing ambient pressure. As in the case of burning gaseous fuel jets, it is assumed that the fuel and oxidant approach each other in stoichiometric proportions. The stoichiometric relations are written as m f  m o i,

m f H  m o Hi  H

(6.99)

where m f and m o are the mass consumption rates per unit volume, H the heat of reaction (combustion) of the fuel per unit mass, and H the heat release rate per unit volume. The mass consumption rates m f and m o are decreasing. There are now three fundamental diffusion equations: one for the fuel, one for the oxidizer, and one for the heat. Equation (6.74) is then written as two equations: one in terms of mf and the other in terms of mo. Equation (6.73) remains the same for the consideration here. As seen in the case of the evaporation, the solution to the equations becomes quite simple when written in terms of the b variable, which led to the Spalding transfer number B. As noted in this case the b variable was obtained from the boundary condition. Indeed, another b variable could have been obtained for the energy equation [Eq. (6.73)]—a variable having the form ⎛ c pT ⎞⎟ ⎟⎟ b  ⎜⎜⎜ ⎜⎝ Lv ⎟⎠ As in the case of burning gaseous fuel jets, the diffusion equations are combined readily by assuming Dρ  (λ/cp), that is, Le  1. The same procedure can be followed in combining the boundary conditions for the three droplet burning equations to determine the appropriate b variables to simplify the solution for the mass consumption rate. The surface boundary condition for the diffusion of fuel is the same as that for pure evaporation [Eq. (6.75)] and takes the form ⎛ dm ρs vs (mfs  1)  Dρ ⎜⎜ f ⎜⎝ dr

⎞⎟ ⎟⎟ ⎟⎠

s

(6.100)

348

Combustion

Since there is no oxidizer leaving the surface, the surface boundary condition for diffusion of oxidizer is ⎛ dm ⎞ 0  ρs vs mos  Dρ ⎜⎜ o ⎟⎟⎟ ⎜⎝ dr ⎟⎠ s

or

⎛ dm ⎞ ρs vs mos  Dρ ⎜⎜ o ⎟⎟⎟ ⎜⎝ dr ⎟⎠ s

(6.101)

The boundary condition for the energy equation is also the same as that would have been for the droplet evaporation case [Eq. (6.75)] and is written as ⎛ λ ⎞⎟ ⎡ d (c T ) ⎤ p ⎜ ⎥ ρs vs Lv  ⎜⎜ ⎟⎟⎟ ⎢⎢ ⎜⎝ c p ⎟⎠ ⎢ dr ⎥⎥ ⎣ ⎦s

(6.102)

Multiplying Eq. (6.100) by H and Eq. (6.101) by iH gives the new forms ⎡ ⎛ dm H[ρs vs (mfs  1)]  ⎢⎢ Dρ ⎜⎜ f ⎜⎝ dr ⎢⎣

⎤ ⎥H ⎥ s ⎥⎦

(6.103)

⎡ ⎧⎪ ⎛ dm ⎞ ⎫⎪ ⎤ H[i{ρs vs mos }]  ⎢⎢ ⎪⎨ Dρ ⎜⎜ o ⎟⎟⎟ ⎪⎬ i ⎥⎥ H ⎜⎝ dr ⎟⎠ ⎪ ⎥ ⎢⎣ ⎪⎪⎩ s⎪ ⎭ ⎦

(6.104)

⎞⎟ ⎟⎟ ⎟⎠

and

By adding Eqs. (6.102) and (6.103) and recalling that Dρ  (λ/cp), one obtains ⎡ d (mf H  c pT ) ⎤ ⎥ ρs vs [ H (mfs  1)  Lv ]  Dρ ⎢⎢ ⎥ dr ⎢⎣ ⎥⎦ s and after transposing, ⎧⎪ ⎪⎪ ⎪⎪ ⎡ m H  c T p f ρs vs  Dρ ⎪⎨ d ⎢⎢ ⎪⎪ ⎢⎣ H (mfs  1)  Lv ⎪⎪ dr ⎪⎪ ⎩

⎫⎪ ⎪⎪ ⎤ ⎪⎪ ⎥ ⎪⎬ ⎥⎪ ⎥⎦ ⎪ ⎪⎪ ⎪⎪ ⎭s

Similarly, by adding Eqs. (6.102) and (6.104), one obtains ⎡d ⎤ ρs vs ( Lv  Himos )  Dρ ⎢ (c pT  mo iH ) ⎥ ⎢⎣ dr ⎥⎦ s

(6.105)

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Diffusion Flames

or ⎡ ⎤ ⎢ ⎥ ⎢ ⎛ m iH  c T ⎞ ⎥ p ⎟⎥ o ⎢ ⎟⎟ ⎥ ρs vs  Dρ ⎢ d ⎜⎜⎜ ⎢ ⎜⎝ Himos  Lv ⎟⎠ ⎥ ⎢ ⎥ ⎢ ⎥ dr ⎣ ⎦s

(6.106)

And, finally, by subtracting Eq. (6.104) from Eq. (6.103), one obtains ⎡ d (mf  imo ) ⎤ ⎥ ρs vs [(mfs  1)  imos ]  Dρ ⎢ ⎢⎣ ⎥⎦ dr s or ⎪⎧⎪ ⎪⎫⎪ ⎪⎪ ⎪ ⎤ ⎪⎪ mf  imo ⎪ ⎡ ⎥ ⎪⎬ ρs vs  Dρ ⎪⎨ d ⎢⎢ ⎪⎪ ⎣ (mfs  1)  imos ⎥⎦ ⎪⎪ ⎪⎪ ⎪⎪ dr ⎪⎪⎩ ⎪⎪⎭ s

(6.107)

Thus, the new b variables are defined as

bfq 

mf H  c pT

H (mfs  1)  Lv mf  imo bfo  (mfs  1)  imos

,

boq 

mo iH  c pT Himos  Lv

, (6.108)

The denominator of each of these three b variables is a constant. The three diffusion equations are transformed readily in terms of these variables by multiplying the fuel diffusion equation by H and the oxygen diffusion equation by iH. By using the stoichiometric relations [Eq. (6.99)] and combining the equations in the same manner as the boundary conditions, one can eliminate the non Again, it is assumed that Dρ  (λ /cp). homogeneous terms m f , m o , and H. The combined equations are then divided by the appropriate denominators from the b variables so that all equations become similar in form. Explicitly then, one has the following developments:

r 2ρv

d (c p T ) dr



d dr

⎡ λ d (c T ) ⎤ p ⎢r2 ⎥  r 2 H ⎢ c ⎥ dr ⎢⎣ ⎥⎦ p

(6.109)

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Combustion

⎪⎧ ⎪⎫ dmf dmf ⎤ d ⎡ 2 ⎢ r Dρ ⎥  r 2 m f ⎪⎬ H ⎪⎨r 2 ρ v  ⎢ ⎥ ⎪⎪⎩ ⎪⎪⎭ dr dr ⎣ dr ⎦

(6.110)

⎪⎧ ⎪⎫ dmo dmo ⎤ d ⎡ 2 ⎢ r Dρ ⎥  r 2 m o ⎪⎬ Hi ⎪⎨r 2 ρ v  ⎪⎪ ⎪⎪ dr dr ⎢⎣ dr ⎥⎦ ⎩ ⎭

(6.111)

m f H  m o Hi  H

(6.99)

Adding Eqs. (6.109) and (6.110), dividing the resultant equation through by [H(mfs  1)  Lv], and recalling that m f H  H , one obtains

r 2ρv

d ⎛⎜ mf H  c pT ⎞⎟⎟ d ⎡⎢ 2 d ⎛⎜ mf H  c pT ⎞⎟⎟⎤⎥ ⎜⎜ ⎜ r D ρ ⎟⎟  ⎟ dr ⎜⎝ H (mfs  1)  Lv ⎠ dr ⎢⎢ dr ⎜⎜⎝ H (mfs  1)  Lv ⎟⎠⎥⎥ ⎣ ⎦

(6.112)

which is then written as ⎛ dbfq ⎞⎟ ⎛ d ⎞ ⎡ ⎛ db ⎞⎤ ⎟⎟  ⎜⎜ ⎟⎟ ⎢⎢ r 2 Dρ ⎜⎜ fq ⎟⎟⎟⎥⎥ r 2 ρ v ⎜⎜⎜ ⎜⎜ dr ⎟ ⎜⎝ dr ⎟⎠ ⎜⎝ dr ⎟⎠ ⎢ ⎝ ⎠⎥⎦ ⎣

(6.113)

Similarly, by adding Eqs. (6.109) and (6.111) and dividing the resultant equation through by [Himos  Lv], one obtains

r 2ρv

d ⎛⎜ mo iH  c pT ⎞⎟⎟ d ⎡⎢ 2 d ⎛⎜ mo iH  c pT ⎞⎟⎟⎤⎥ ⎜⎜ ⎜ r Dρ ⎟⎟  ⎟ ⎢ dr ⎜⎝ Himos  Lv ⎠ dr ⎢ dr ⎜⎜⎝ Himos  Lv ⎟⎠⎥⎥ ⎣ ⎦

(6.114)

⎤ d d ⎡ 2 d ⎢ r Dρ (boq )  (boq ) ⎥ ⎢ ⎥⎦ dr dr ⎣ dr

(6.115)

or

r 2ρv

Following the same procedures by subtracting Eq. (6.111) from Eq. (6.110), one obtains

r 2ρv

⎤ d d ⎡ 2 d ⎢ r Dρ (bfo )  (bfo ) ⎥ ⎥⎦ dr dr ⎢⎣ dr

(6.116)

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Diffusion Flames

Obviously, all the combined equations have the same form and boundary conditions, that is,

r 2ρv

⎡ 2 ⎤ d ⎢ r Dρ (b)fo,fq,oq ⎥ ⎢⎣ ⎥⎦ dr ⎡d ⎤ ρs vs  Dρ ⎢ (b)fo,fq,oq ⎥ at r  rs ⎢⎣ dr ⎥⎦ s b  b at r → 

d d (b)fo,fq,oq  dr dr

(6.117)

The equation and boundary conditions are the same as those obtained for the pure evaporation problem; consequently, the solution is the same. Thus, one writes

Gf 

⎛ D ρ ⎞⎟ m f ⎜⎜ s s ⎟ ln(1  B) where B  b  b   s ⎜⎜ r ⎟⎟ 4πrs2 ⎝ s ⎠

(6.118)

It should be recognized that since Le  Sc  Pr, Eq. (6.118) can also be written as ⎛ λ ⎞⎟ ⎛ ⎞ ⎜ ⎟⎟ ln(1  B)  ⎜⎜ μ ⎟⎟⎟ ln(1  B) Gf  ⎜⎜ ⎜⎜ r ⎟ ⎜⎝ c p rs ⎟⎟⎠ ⎝ s⎠

(6.119)

In Eq. (6.118) the transfer number B can take any of the following forms: Without combustion assumption

With combustion assumption

(m  mfs )  (mos  mo )i im  mfs Bfo  f  o (mfs  1)  imos 1  mfs H (mf  mfs )  c p (T  Ts ) c p (T  Ts )  mfs H Bfq   Lv  H (mfs  1) Lv  H (mfs  1) Hi(mo  mos )  c p (T  Ts ) c p (T  Ts )  imo H Boq   Lv  imos H Lv

(6.120)

The combustion assumption in Eq. (6.120) is that mos  mf  0 since it is assumed that neither fuel nor oxidizer can penetrate the flame zone. This requirement is not that the flame zone be infinitely thin, but that all the oxidizer must be consumed before it reaches the fuel surface and that the quiescent atmosphere contain no fuel.

352

Combustion

As in the evaporation case, in order to solve Eq. (6.118), it is necessary to proceed by first equating Bfo  Boq. This expression c p (T  Ts )  imo H imo  mfs  1  mfs Lv

(6.121)

is solved with the use of the vapor pressure data for the fuel. The iteration process described in the solution of BM  BT in the evaporation problem is used. The solution of Eq. (6.121) gives Ts and mfs and, thus, individually Bfo and Boq. With B known, the burning rate is obtained from Eq. (6.118). For the combustion systems, Boq is the most convenient form of B: cp(T  Ts) is usually much less than imo H and to a close approximation Boq  (imo  H/Lv). Thus the burning rate (and, as will be shown later, the evaporation coefficient β) is readily determined. It is not necessary to solve for mfs and Ts. Furthermore, detailed calculations reveal that Ts is very close to the saturation temperature (boiling point) of most liquids at the given pressure. Thus, although Dρ in the burning rate expression is independent of pressure, the transfer number increases as the pressure is raised because pressure rises concomitantly with the droplet saturation temperature. As the saturation temperature increases, the latent heat of evaporation Lv decreases. Of course, Lv approaches zero at the critical point. This increase in Lv is greater than the decrease of the cpg(T  Ts) and cpl(Ts  Tl) terms; thus B increases with pressure and the burning follows some logarithmic trend through ln(1  B). The form of Boq and Bfq presented in Eq. (6.120) is based on the assumption that the fuel droplet has infinite thermal conductivity, that is, the temperature of the droplet is Ts throughout. But in an actual porous sphere experiment, the fuel enters the center of the sphere at some temperature Tl and reaches Ts at the sphere surface. For a large sphere, the enthalpy required to raise the cool entering liquid to the surface temperature is cpl(Ts  Tl) where cpl is the specific heat of the liquid fuel. To obtain an estimate of B that gives a conservative (lower) result of the burning rate for this type of condition, one could replace Lv by Lv  Lv  c pl (Ts  Tl )

(6.122)

in the forms of B represented by Eq. (6.120). Table 6.1, extracted from Kanury [19], lists various values of B for many condensed phase combustible substances burning in the air. Examination of this table reveals that the variation of B for different combustible liquids is not great; rarely does one liquid fuel have a value of B a factor of 2 greater than another. Since the transfer number always enters the burning rate expression as a ln(1  B) term, one may conclude that, as long as the oxidizing atmosphere is kept the same, neither the burning rate nor the evaporation coefficient of liquid fuels will vary greatly. The diffusivities and gas density dominate the burning

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TABLE 6.1 Transfer Numbers of Various Liquids in Aira B

B

B

B 

n-Pentane

8.94

8.15

8.19

9.00

n-Hexane

8.76

6.70

6.82

8.83

n-Heptane

8.56

5.82

6.00

8.84

n-Octane

8.59

5.24

5.46

8.97

i-Octane

9.43

5.56

5.82

9.84

n-Decane

8.40

4.34

4.62

8.95

n-Dodecane

8.40

4.00

4.30

9.05

Octene

9.33

5.64

5.86

9.72

Benzene

7.47

6.05

6.18

7.65

Methanol

2.95

2.70

2.74

3.00

Ethanol

3.79

3.25

3.34

3.88

Gasoline

9.03

4.98

5.25

9.55

Kerosene

9.78

3.86

4.26

10.80

Light diesel

10.39

3.96

4.40

11.50

Medium diesel

11.18

3.94

4.38

12.45

Heavy diesel

11.60

3.91

4.40

13.00

Acetone

6.70

5.10

5.19

6.16

Toluene

8.59

6.06

6.30

8.92

Xylene

9.05

5.76

6.04

9.48

Note. B  [im∞H  c p (T∞  Ts)]/ L v;  B ′[im∞H  c p (T∞  Ts)]/ L ′v ; B ′′  (imo∞H / L ′v ); B ′′′  (imo∞H / L v )

a

T  20°C.

rate. Whereas a tenfold variation in B results in an approximately twofold variation in burning rate, a tenfold variation of the diffusivity or gas density results in a tenfold variation in burning. Furthermore, note that the burning rate has been determined without determining the flame temperature or the position of the flame. In the approach attributed to Godsave [20] and extended by others [21, 22], it was necessary to find the flame temperature, and the early burning rate developments largely followed this procedure. The early literature contains frequent comparisons not only of the calculated and experimental burning rates (or β), but also of the

354

Combustion

flame temperature and position. To their surprise, most experimenters found good agreement with respect to burning rate but poorer agreement in flame temperature and position. What they failed to realize is that, as shown by the discussion here, the burning rate is independent of the flame temperature. As long as an integrated approach is used and the gradients of temperature and product concentration are zero at the outer boundary, it does not matter what the reactions are or where they take place, provided they take place within the boundaries of the integration. It is possible to determine the flame temperature Tf and position rf corresponding to the Godsave-type calculations simply by assuming that the flame exists at the position where imo  mf. Equation (6.85) is written in terms of bfo as ⎡ m  m  i(m  m )  (m  1)  im ⎤ rs2 ρs vs fs o os fs os ⎥  ln ⎢⎢ f ⎥ Ds ρs r m  m  i ( m  m )  ( m  1 )  im f fs o os fs os ⎣ ⎦

(6.123)

At the flame surface, mf  mo  0 and mf  mos  0; thus Eq. (6.123) becomes rs2 ρs vs  ln(1  imo ) Ds ρs rf

(6.124)

Since m  4πrs2 ρs vs is known, rf can be estimated. Combining Eqs. (6.118) and (6.124) results in the ratio of the radii rf ln(1  B)  rs ln(1  imo )

(6.125)

For the case of benzene (C6H6) burning in air, the mass stoichiometric index i is found to be i

78  0.325 7.5  32

Since the mass fraction of oxygen in air is 0.23 and B from Table 6.1 is given as 6, one has rf ln(1  6)   27 ln[1  (0.325  0.23)] rs This value is much larger than the value of about 2 to 4 observed experimentally. The large deviation between the estimated value and that observed is

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Diffusion Flames

most likely due to the assumptions made with respect to the thermophysical properties and the Lewis number. This point is discussed in Chapter 6, Section C2c. Although this estimate does not appear suitable, it is necessary to emphasize that the results obtained for the burning rate and the combustion evaporation coefficient derived later give good comparisons with experimental data. The reason for this supposed anomaly is that the burning rate is obtained by integration between the infinite atmosphere and the droplet and is essentially independent of the thermophysical parameters necessary to estimate internal properties. Such parameters include the flame radius and the flame temperature, to be discussed in subsequent paragraphs. For surface burning, as in an idealized case of char burning, it is appropriate to assume that the flame is at the particle surface and that rf  rs. Thus from Eq. (6.125), B must equal imo. Actually, the same result would arise from the first transfer number in Eq. (6.120)

Bfo 

imo  mfs 1  mfs

Under the condition of surface burning, mfs  0 and thus, again, Bfo  imo. As in the preceding approach, an estimate of the flame temperature Tf at rf can be obtained by writing Eq. (6.85) with b  boq. For the condition that Eq. (6.85) is determined at r  rf, one makes use of Eq. (6.124) to obtain the expression ⎡ b  b 1⎤ s ⎥ 1  imo  ⎢⎢  ⎥ b b 1   f s ⎣ ⎦ [ Himo  c p (T  Ts )  Lv ] = [c p (Tf  Ts )  Lv ]

(6.126)

or ⎧⎪ [ Himo  c p (T  Ts )  Lv ] ⎫⎪ ⎪⎬  L c p (T  Ts )  ⎪⎨ v ⎪⎪ ⎪⎪ 1  im o  ⎩ ⎭ Again, comparisons of results obtained by Eq. (6.126) with experimental measurements show the same extent of deviation as that obtained for the determination of rf/rs from Eq. (6.125). The reasons for this deviation are also the same. Irrespective of these deviations, it is also possible from this transfer number approach to obtain some idea of the species profiles in the droplet burning case. It is best to establish the conditions for the nitrogen profile for this

356

Combustion

quasi-steady fuel droplet burning case in air first. The conservation equation for the inert i (nitrogen) in droplet burning is 4πr 2 ρ vmi  4πr 2 Dρ

dmi dr

(6.127)

Integrating for the boundary condition that as r → , mi  mi, one obtains 

⎛ m ⎞ 4πrs2 ρs vs 1  Dρ ln ⎜⎜⎜ i ⎟⎟⎟ ⎜⎝ mi ⎟⎠ 4π r

(6.128)

From either Eq. (6.86) or Eq. (6.118) it can be seen that rs ρs vs  Ds ρs ln(1  B) Then Eq. (6.128) can be written in the form rs ρs vs

⎛ m ⎞ rs r   ⎡⎣ Ds ρs ln(1  B) ⎤⎦ s  Dρ ln ⎜⎜⎜ i ⎟⎟⎟ ⎜⎝ mi ⎟⎠ r r

(6.129)

For the condition at r  rs, this last expression reveals that ⎛m ⎞  ln(1  B)  ln ⎜⎜⎜ is ⎟⎟⎟ ⎜⎝ mi ⎟⎠ or 1 B 

mi  mis

Since the mass fraction of nitrogen in air is 0.77, for the condition B  6 mis 

0.77  0.11 7

Thus the mass fraction of the inert nitrogen at the droplet surface is 0.11. Evaluating Eq. (6.129) at r  rf, one obtains 

⎛m ⎞ rs ln(1  B)  ln ⎜⎜⎜ if ⎟⎟⎟ ⎜⎝ mi ⎟⎠ rf

357

Diffusion Flames

However, Eq. (6.125) gives rs ln(1  B)  ln(1  imo ) rf Thus 1  imo 

mi  mif

or for the case under consideration mif 

0.77  0.72 1.075

The same approach would hold for the surface burning of a carbon char except that i would equal 0.75 (see Chapter 9) and mis 

0.77  0.66 1.075  0.22

which indicates that the mass fraction of gaseous products mps for this carbon case equals 0.34. For the case of a benzene droplet in which B is taken as 6, the expression that permits the determination of mfs is B

imo  mfs 6 1  mfs

or mfs  0.85. This transfer number approach for a benzene droplet burning in air reveals species profiles characteristic of most hydrocarbons and gives the results summarized below: mfs mos mi s mps

 0.85 mff  0.00 mof  0.11 mif  0.04 mpf

 0.00 mf  0.00 mo  0.72 mi  0.28 mp

 0.00  0.23  0.77  0.00

These results are graphically represented in Fig. 6.11. The product composition in each case is determined for the condition that the mass fractions at each position must total to 1.

358

Combustion

0.025 d 2 (cm2) 0.020 drop diam

Benzene

2

0.015 0.010

0

0.5 t (sec)

1.0

FIGURE 6.13 Diameter–time measurements of a benzene droplet burning in quiescent air showing diameter-squared dependence (after Godsave [20]).

It is now possible to calculate the burning rate of a droplet under the quasisteady conditions outlined and to estimate, as well, the flame temperature and position; however, the only means to estimate the burning time of an actual droplet is to calculate the evaporation coefficient for burning, β. From the mass burning results obtained, β may be readily determined. For a liquid droplet, the relation ⎛ dr ⎞ dm  m f  4πρl rs2 ⎜⎜ s ⎟⎟⎟ ⎜⎝ dt ⎟⎠ dt

(6.130)

gives the mass rate in terms of the rate of change of radius with time. Here, ρl is the density of the liquid fuel. It should be evaluated at Ts. Many experimenters have obtained results similar to those given in Fig. 6.13. These results confirm that the variation of droplet diameter during burning follows the same “law” as that for evaporation. d 2  do2  β t

(6.131)

drs β  8rs dt

(6.132)

Since d  2rs,

It is readily shown that Eqs. (6.118) and (6.130) verify that a “d2” law should exist. Equation (6.130) is rewritten as ⎛ dr 2 ⎞ ⎛ πρ r ⎞ ⎡ d (d 2 ) ⎤ ⎥ m f  2πρl rs ⎜⎜⎜ s ⎟⎟⎟   ⎜⎜ l s ⎟⎟⎟ ⎢ ⎜⎝ 2 ⎟⎠ ⎢ dt ⎥ ⎜⎝ dt ⎟⎠ ⎣ ⎦

359

Diffusion Flames

Making use of Eq. (6.118) ⎛ λ ⎞⎟ ⎛ρ m f ⎜  ⎜⎜ ⎟⎟⎟ ln(1  B)   ⎜⎜ l ⎜⎝ 8 ⎜⎝ c p ⎟⎠ 4πrs

⎛ρ ⎞⎟ ⎡ d (d 2 ) ⎤ ⎥   ⎜⎜ l ⎟⎟ ⎢ ⎜⎝ 8 ⎟⎠ ⎢ dt ⎥ ⎦ ⎣

⎞⎟ ⎟⎟ β ⎟⎠

shows that d (d 2 )  const, dt

⎛8 β  ⎜⎜⎜ ⎜⎝ ρl

⎞⎟ ⎛⎜ λ ⎞⎟ ⎟⎟ ⎜⎜ ⎟⎟ ln(1  B) ⎠⎟ ⎜⎝ c p ⎟⎟⎠

(6.133)

which is a convenient form since λ/cp is temperature-insensitive. Sometimes β is written as ⎛ρ ⎞ ⎛ ρ D ⎞⎟ ⎟ ln(1  B) β  8 ⎜⎜⎜ s ⎟⎟⎟ α ln(1  B)  8 ⎜⎜⎜ ⎜⎝ ρl ⎟⎠ ⎜⎝ ρl ⎟⎟⎠ to correspond more closely to expressions given by Eqs. (6.96) and (6.118). The proper response to Problem 9 of this chapter will reveal that the “d2” law is not applicable in modest Reynolds number or convective flow.

c. Refinements of the Mass Burning Rate Expression Some major assumptions were made in the derivation of Eqs. (6.118) and (6.133). First, it was assumed that the specific heat, thermal conductivity, and the product Dρ were constants and that the Lewis number was equal to 1. Second, it was assumed that there was no transient heating of the droplet. Furthermore, the role of finite reaction kinetics was not addressed adequately. These points will be given consideration in this section. Variation of Transport Properties The transport properties used throughout the previous developments are normally strong functions of both temperature and species concentration. The temperatures in a droplet diffusion flame as depicted in Fig. 6.11 vary from a few hundred degrees Kelvin to a few thousand. In the regions of the droplet flame one essentially has fuel, nitrogen, and products. However, under steady-state conditions as just shown the major constituent in the sf region is the fuel. It is most appropriate to use the specific heat of the fuel and its binary diffusion coefficient. In the region f, the constituents are oxygen, nitrogen, and products. With similar reasoning, one essentially considers the properties of nitrogen to be dominant; moreover, as the properties of fuel are used in the sf region, the properties of nitrogen are appropriate to be used in the f∞ region. To illustrate the importance of variable properties, Law [23] presented a simplified model in which the temperature dependence was suppressed, but the concentration dependence was represented

360

Combustion

by using λ, cp, and Dρ with constant, but different, values in the sf and f regions. The burning rate result of this model was [λ / ( c p )sf ] ⎧⎪ ⎡ ⎫⎪ c pf (Tf  Ts ) ⎤ ⎪⎪ ⎢ ⎪⎪ m ( ρ ) D ⎥ f   ln ⎨ ⎢1  [1  imo ] ⎬ ⎥ ⎪ ⎪⎪ 4πrs Lv ⎥⎦ ⎪⎪ ⎢⎣ ⎪⎭ ⎩

(6.134)

where Tf is obtained from expressions similar to Eq. (6.126) and found to be

(c p )sf (Tf  Ts ) 

(c p )f (Tf  T ) [(1  imo )1/ ( Le)f  1]

 H  Lv

(6.135)

(rf/rs) was found to be ⎡ (λ /c p )sf ln[1  (c p )fs (Tf  Ts ) /Lv ] ⎤ rf ⎥  1  ⎢⎢ ⎥ rs ln(1  imo ) ⎥⎦ ⎢⎣ ( Dρ )f

Law [23] points out that since imo is generally much less than 1, the denominator of the second term in Eq. (6.135) becomes [imo/(Le)f], which indicates that the effect of (Le)f is to change the oxygen concentration by a factor ( Le)1 f as experienced by the flame. Obviously, then, for (Le)f 1, the oxidizer concentration is effectively reduced and the flame temperature is also reduced from the adiabatic value obtained for Le  1, the value given by Eq. (6.126). The effective Lewis number for the mass burning rate [Eq. (6.134)] is ⎛ λ ⎞⎟ ⎜ Le  ⎜⎜ ⎟⎟⎟ ( Dρ )f ⎜⎝ c p ⎟⎠ sf which reveals that the individual Lewis numbers in the two regions play no  Law and Law [24] estimated that for heptane burnrole in determining m. ing in air the effective Lewis number was between 1/3 and 1/2. Such values in Eq. (6.136) predict values of rf/rs in the right range of experimental values [23]. The question remains as to the temperature at which to evaluate the physical properties discussed. One could use an arithmetic mean for the two regions [25] or Sparrow’s 1/3 rule [26], which gives Tsf  13 Ts  23 Tf ;

Tf  13 Tf  23 T

361

Diffusion Flames

Transient Heating of Droplets When a cold liquid fuel droplet is injected into a hot stream or ignited by some other source, it must be heated to its steady-state temperature Ts derived in the last section. Since the heat-up time can influence the “d2” law, particularly for high-boiling-point fuels, it is of interest to examine the effect of the droplet heating mode on the main bulk combustion characteristic—the burning time. For this case, the boundary condition corresponding to Eq. (6.102) becomes ⎛ ⎛ ∂T ⎞⎟ ∂T ⎞⎟  v  ⎜⎜ 4πr 2λl  v mL ⎟  mL ⎟⎟  ⎜⎜ 4πr 2λg ⎜⎝ ∂r ⎟⎠s ∂r ⎠s ⎜⎝

(6.136)

where the subscript s designates the liquid side of the liquid–surface–gas interface and s designates the gas side. The subscripts l and g refer to liquid and gas, respectively. At the initiation of combustion, the heat-up (second) term of Eq. (6.136) can be substantially larger than the vaporization (first) term. Throughout combustion the third term is fixed. Thus, some [27, 28] have postulated that droplet combustion can be considered to consist of two periods: namely, an initial droplet heating period of slow vaporization with ⎛ ∂T ⎞⎟  v  ⎜⎜ 4πr 2λl mL ⎟ ⎜⎝ ∂r ⎟⎠s followed by fast vaporization with almost no droplet heating so that  v  mL  v mL The extent of the droplet heating time depends on the mode of ignition and the fuel volatility. If a spark is used, the droplet heating time can be a reasonable percentage (10–20%) of the total burning time [26]. Indeed, the burning time would be 10–20% longer if the value of B used to calculate β is calculated with Lv  Lv, that is, on the basis of the latent heat of vaporization. If the droplet is ignited by injection into a heated gas atmosphere, a long heating time will precede ignition; and after ignition, the droplet will be near its saturation temperature so the heat-up time after ignition contributes little to the total burn-up time. To study the effects due to droplet heating, one must determine the temperature distribution T(r, t) within the droplet. In the absence of any internal motion, the unsteady heat transfer process within the droplet is simply described by the heat conduction equation and its boundary conditions α ∂ ⎛⎜ 2 ∂T ⎞⎟ ∂T  2l ⎟, ⎜r r ∂r ⎜⎝ ∂r ⎟⎠ ∂t

T (r , t  0)  Tl (r ),

⎛ ∂T ⎞⎟ ⎜⎜ ⎟ ⎝⎜ ∂r ⎟⎠

r0

0

(6.137)

362

Combustion

The solution of Eq. (6.137) must be combined with the nonsteady equations for the diffusion of heat and mass. This system can only be solved numerically and the computing time is substantial. Therefore, a simpler alternative model of droplet heating is adopted [26, 27]. In this model, the droplet temperature is assumed to be spatially uniform at Ts and temporally varying. With this assumption Eq. (6.136) becomes ⎛ ⎛4⎞ ∂Ts ∂T ⎞⎟  v  ⎜⎜ 4πrs2λg  v  ⎜⎜ ⎟⎟⎟ πrs3ρl (c p )l mL ⎟⎟  mL ⎜⎝ ⎜⎝ 3 ⎠ ∂r ⎠s ∂t

(6.138)

But since ⎤ ⎛ d ⎞ ⎡⎛ 4 ⎞ m   ⎜⎜ ⎟⎟⎟ ⎢⎜⎜ ⎟⎟⎟ πrs3ρl ⎥ ⎥ ⎜⎝ dt ⎠ ⎢⎜⎝ 3 ⎠ ⎣ ⎦ Eq. (6.138) integrates to Ts ⎪⎧ ⎛ r ⎞⎟3 ⎛ dT ⎞⎟⎫⎪⎪ s ⎜⎜ s ⎟  exp ⎪⎪c ⎜ ⎟⎪ ⎨ pl ∫ ⎜ ⎜⎜ r ⎟⎟ ⎜⎜ L  L ⎟⎟⎬⎪ ⎪ ⎝ so ⎠ v ⎠⎪ ⎪⎪⎩ Tso ⎝ v ⎪⎭

(6.139)

where Lv  Lv (Ts ) is given by Lv 

(1  mfs )[imo H  c p (T  Ts )]

(6.140)

mfs  imo

and rso is the original droplet radius. Figure 6.14, taken from Law [28], is a plot of the nondimensional radius squared as a function of a nondimensional time for an octane droplet initially at 300 K burning under ambient conditions. Shown in this figure are the droplet histories calculated using Eqs. (6.137) and (6.138). Sirignano and Law [27] refer to the result of Eq. (6.137) as the diffusion limit and that of Eq. (6.138) as the distillation limit, respectively. Equation (6.137) allows for diffusion of heat into the droplet, whereas Eq. (6.138) essentially assumes infinite thermal conductivity of the liquid and has vaporization at Ts as a function of time. Thus, one should expect Eq. (6.137) to give a slower burning time. Also plotted in Fig. 6.14 are the results from using Eq. (6.133) in which the transfer number was calculated with Lv  Lv and Lv  Lv  cpl(Ts  Tl) [Eq. (6.122)]. As one would expect, Lv  Lv gives the shortest burning time, and Eq. (6.137) gives the longest burning time since it does not allow storage of heat in the droplet as it burns. All four results are remarkably close for octane. The spread could be greater for a heavier fuel. For a conservative estimate of the burning time, use of B with Lv evaluated by Eq. (6.122) is recommended. The Effect of Finite Reaction Rates When the fuel and oxidizer react at a finite rate, the flame front can no longer be considered infinitely thin. The reaction

363

Diffusion Flames

1.0 d 2 —Law model for B Diffusion—limit model Rapid—mixing model d 2 —Law model for B1

(R/Ro)2

0.8

0.6

0.4

0.2

0

0

0.1 2 (αᐉ/RO )

0.2 t

FIGURE 6.14 Variation of nondimensional droplet radius as a function of nondimensional time during burning with droplet heating and steady-state models (after Law [28]).

rate is then such that oxidizer and fuel can diffuse through each other and the reaction zone is spread over some distance. However, one must realize that although the reaction rates are considered finite, the characteristic time for the reaction is also considered to be much shorter than the characteristic time for the diffusional processes, particularly the diffusion of heat from the reaction zone to the droplet surface. The development of the mass burning rate [Eq. (6.118)] in terms of the transfer number B [Eq. (6.120)] was made with the assumption that no oxygen reaches the fuel surface and no fuel reaches , the ambient atmosphere. In essence, the only assumption made was that the chemical reactions in the gasphase flame zone were fast enough so that the conditions mos  0  mf could be met. The beauty of the transfer number approach, given that the kinetics are finite but faster than diffusion and the Lewis number is equal to 1, is its great simplicity compared to other endeavors [20, 21]. For infinitely fast kinetics, then, the temperature profiles form a discontinuity at the infinitely thin reaction zone (see Fig. 6.11). Realizing that the mass burning rate must remain the same for either infinite or finite reaction rates, one must consider three aspects dictated by physical insight when the kinetics are finite: first, the temperature gradient at r  rs must be the same in both cases; second, the maximum temperature reached when the kinetics are finite must be less than that for the infinite kinetics case; third, if the temperature is lower in the finite case, the maximum must be closer to the droplet in order to satisfy the first aspect. Lorell et al. [22] have shown analytically that these physical insights as depicted in Fig. 6.15 are correct.

364

0.05 0.04 0.03 0.02 0.01 0

1.2 T3

1.0

T2 T1

0.8

0.32 Z2

0.6 Z1 0.4

mO1

0.2 0 0.128

0.132

0.31 mO3 m mF3 O2 mF2

0.136 0.140 Position r (cm)

mF 1 0.144

Dimensionless temperature (T°)

Weight fractions of mf and oxidizer mO

0.06

Fraction of reaction completed z

Combustion

0.30

FIGURE 6.15 Effect of chemical rate processes on the structure of a diffusion-controlled droplet flame (after Lorell et al. [22]).

D. BURNING OF DROPLET CLOUDS Current understanding of how particle clouds and sprays burn is still limited, despite numerous studies—both analytical and experimental—of burning droplet arrays. The main consideration in most studies has been the effect of droplet separation on the overall burning rate. It is questionable whether study of simple arrays will yield much insight into the burning of particle clouds or sprays. An interesting approach to the spray problem has been suggested by Chiu and Liu [29], who consider a quasi-steady vaporization and diffusion process with infinite reaction kinetics. They show the importance of a group combustion number (G), which is derived from extensive mathematical analyses and takes the form ⎛R⎞ G  3(1  0.276 Re1/ 2 Sc1/ 2 Le N 2 / 3 ) ⎜⎜ ⎟⎟⎟ ⎜⎝ S ⎠

(6.141)

where Re, Sc, and Le are the Reynolds, Schmidt, and Lewis numbers, respectively; N the total number of droplets in the cloud; R the instantaneous average radius; and S the average spacing of the droplets. The value of G was shown to have a profound effect upon the flame location and distribution of temperature, fuel vapor, and oxygen. Four types of behaviors were found for large G numbers. External sheath combustion occurs for the largest value; and as G is decreased, there is external group combustion, internal group combustion, and isolated droplet combustion. Isolated droplet combustion obviously is the condition for a separate flame envelope for each droplet. Typically, a group number less that 102 is required. Internal group combustion involves a core with a cloud where vaporization exists such that the core is totally surrounded by flame. This condition occurs

365

Diffusion Flames

mO,∞

δ

Flame y

Condensed phase fuel FIGURE 6.16 Stagnant film height representation for condensed phase burning.

for G values above 102 and somewhere below 1. As G increases, the size of the core increases. When a single flame envelops all droplets, external group combustion exists. This phenomenon begins with G values close to unity. (Note that many industrial burners and most gas turbine combustors are in this range.) With external group combustion, the vaporization of individual droplets increases with distance from the center of the core. At very high G values (above 102), only droplets in a thin layer at the edge of the cloud vaporize. This regime is called the external sheath condition.

E. BURNING IN CONVECTIVE ATMOSPHERES 1. The Stagnant Film Case The spherical-symmetric fuel droplet burning problem is the only quiescent case that is mathematically tractable. However, the equations for mass burning may be readily solved in one-dimensional form for what may be considered the stagnant film case. If the stagnant film is of thickness δ, the freestream conditions are thought to exist at some distance δ from the fuel surface (Fig. 6.16). Within the stagnant film, the energy equation can be written as ⎛ dT ⎞⎟ ⎛ d 2T ⎞⎟ ⎟⎟  λ ⎜⎜ ⎟  ρ vc p ⎜⎜ ⎜⎝ dy 2 ⎟⎟⎠  H ⎜⎝ dy ⎟⎠

(6.142)

With b defined as before, the solution of this equation and case proceeds as follows. Analogous to Eq. (6.117), for the one-dimensional case ⎛ db ⎞ ⎛ d2b ⎞ ρ v ⎜⎜⎜ ⎟⎟⎟  ρ D ⎜⎜⎜ 2 ⎟⎟⎟ ⎝ dy ⎟⎠ ⎝ dy ⎟⎠

(6.143)

366

Combustion

Integrating Eq. (6.143), one has db  const dy

(6.144)

⎛ db ⎞ ρ D ⎜⎜⎜ ⎟⎟⎟  ρs vs  ρ v ⎝ dy ⎟⎠o

(6.145)

ρ vb = ρ D The boundary condition is

Substituting this boundary condition into Eq. (6.144), one obtains ρ vbo  ρ v  const,

ρ v(bo  1)  const

The integrated equation now becomes ρ v(b − bo + 1) = ρ D

db dy

(6.146)

which upon second integration becomes ρ vy  ρ D ln(b  bo  1)  const

(6.147)

At y  0, b  bo; therefore, the constant equals zero so that ρ vy  ρ D ln(b  bo  1)

(6.148)

Since δ is the stagnant film thickness, ρ vδ  ρ D ln(bδ  bo  1)

(6.149)

⎛ ρ D ⎞⎟ Gf  ⎜⎜ ⎟ ln(1  B) ⎜⎝ δ ⎟⎠

(6.150)

B  bδor  bo

(6.151)

Thus,

where, as before

Since the Prandtl number cp μ /λ is equal to 1, Eq. (6.150) may be written as ⎛ λ ⎞⎟ ⎛ ⎞ ⎛ ρ D ⎞⎟ ⎜ ⎟⎟ ln(1  B)  ⎜⎜ μ ⎟⎟ ln(1  B) Gf  ⎜⎜ ⎟⎟ ln(1  B)  ⎜⎜ ⎟ ⎜⎝ δ ⎟⎠ ⎜⎝ δ ⎠ ⎜⎝ c pδ ⎟⎠

(6.152)

367

Diffusion Flames

A burning pool of liquid or a volatile solid fuel will establish a stagnant film height due to the natural convection that ensues. From analogies to heat transfer without mass transfer, a first approximation to the liquid pool burning rate may be written as Gf d  Gr a μ ln(1  B)

(6.153)

where a equals 1/4 for laminar conditions and 1/3 for turbulent conditions, d the critical dimension of the pool, and Gr the Grashof number: ⎛ gd 3 β ⎞ Gr  ⎜⎜⎜ 2 1 ⎟⎟⎟ ΔT ⎜⎝ α ⎟⎠

(6.154)

where g is the gravitational constant and β1 the coefficient of expansion. When the free stream—be it forced or due to buoyancy effects—is transverse to the mass evolution from the regressing fuel surface, no stagnant film forms, in which case the correlation given by Eq. (6.153) is not explicitly correct.

2. The Longitudinally Burning Surface Many practical cases of burning in convective atmospheres may be approximated by considering a burning longitudinal surface. This problem is similar to what could be called the burning flat plate case and has application to the problems that arise in the hybrid rocket. It differs from the stagnant film case in that it involves gradients in the x direction as well as the y direction. This configuration is depicted in Fig. 6.17. For the case in which the Schmidt number is equal to 1, it can be shown [7] that the conservation equations [in terms of Ω; see Eq. (6.17)] can be transposed into the form used for the momentum equation for the boundary layer. Indeed, the transformations are of the same form as the incompressible boundary layer equations developed and solved by Blasius [30]. The important difference

Oxidizer δ

Flame y x

Condensed phase fuel FIGURE 6.17 Burning of a flat fuel surface in a one-dimensional flow field.

368

Combustion

between this problem and the Blasius [30] problem is the boundary condition at the surface. The Blasius equation takes the form f   ff  0

(6.155)

where f is a modified stream function and the primes designate differentiation with respect to a transformed coordinate. The boundary conditions at the surface for the Blasius problem are η  0, η  ,

f (0)  0, f ()  1

and

f (0)  0

(6.156)

For the mass burning problem, the boundary conditions at the surface are η  0, η  ,

f ′(0)  0, f ′(  )  1

and

Bf (0)  0

(6.157)

where η is the transformed distance normal to the plate. The second of these conditions contains the transfer number B and is of the same form as the boundary condition in the stagnant film case [Eq. (6.145)]. Emmons [31] solved this burning problem, and his results can be shown to be of the form ⎛ λ ⎞⎟ ⎛ Re1/ 2 ⎞ ⎜ Gf  ⎜⎜ ⎟⎟⎟ ⎜⎜⎜ x ⎟⎟⎟ [ f (0)] ⎜⎝ c p ⎟⎠ ⎜⎝ x 2 ⎟⎠

(6.158)

where Rex is the Reynolds number based on the distance x from the leading edge of the flat plate. For Prandtl number equal to 1, Eq. (6.158) can be written in the form Re1/ 2 [ f (0)] Gf x  x μ 2

(6.159)

It is particularly important to note that [f(0)] is a function of the transfer number B. This dependence as determined by Emmons is shown in Fig. 6.18. An interesting approximation to this result [Eq. (6.159)] can be made from the stagnant film result of the last section, that is, ⎛ λ ⎞⎟ ⎛ ⎞ ⎛ ρ D ⎞⎟ ⎜ ⎟⎟ ln(1  B)  ⎜⎜ μ ⎟⎟ ln(1  B) Gf  ⎜⎜ ⎟⎟ ln(1  B)  ⎜⎜ ⎟ ⎜⎝ δ ⎟⎠ ⎜⎝ δ ⎠ ⎜⎝ c pδ ⎟⎠

(6.152)

369

Diffusion Flames

1.0

[f(0)]

8 6 4 x

2 0 0

1

In(1B ) B 0.152.6

10

100

B FIGURE 6.18 [f(0)] as a function of the transfer number B.

If δ is assumed to be the Blasius boundary layer thickness δx, then 1/ 2 δx  5.2 x Re x

(6.160)

Substituting Eq. (6.160) into Eq. (6.152) gives Gf x ⎛⎜ Re1x / 2 ⎞⎟ ⎟ ln(1  B)  ⎜⎜ ⎜⎝ 5.2 ⎟⎟⎠ μ

(6.161)

The values predicted by Eq. (6.161) are somewhat high compared to those predicted by Eq. (6.159). If Eq. (6.161) is divided by B0.15/2 to give Gf x ⎛⎜ Re1x/ 2 ⎞⎟ ⎡ ln(1  B) ⎤ ⎟⎢ ⎥  ⎜⎜ ⎜⎝ 2.6 ⎟⎟⎠ ⎢⎣ B0.15 ⎥⎦ μ

(6.162)

the agreement is very good over a wide range of B values. To show the extent of the agreement, the function ln(1  B) B0.15 2.6

(6.163)

is plotted on Fig. 6.18 as well. Obviously, these results do not hold at very low Reynolds numbers. As Re approaches zero, the boundary layer thickness approaches infinity. However, the burning rate is bounded by the quiescent results.

3. The Flowing Droplet Case When droplets are not at rest relative to the oxidizing atmosphere, the quiescent results no longer hold, so forced convection must again be considered. No one

370

Combustion

has solved this complex case. As discussed in Chapter 4, Section E, flow around a sphere can be complex, and at relatively high Re( 20), there is a boundary layer flow around the front of the sphere and a wake or eddy region behind it. For this burning droplet case, an overall heat transfer relationship could be written to define the boundary condition given by Eq. (6.90). h(ΔT )  Gf Lv

(6.164)

The thermal driving force is represented by a temperature gradient ΔT which is the ambient temperature T plus the rise in this temperature due to the energy release minus the temperature at the surface Ts, or ⎛ im H ⎞⎟ ⎡ im H  c (T  T ) ⎤ o p  s ⎥ ⎜ ΔT  T  ⎜⎜ o ⎟⎟⎟  Ts  ⎢⎢ ⎥ ⎜⎝ c p ⎟⎠ cp ⎢⎣ ⎥⎦

(6.165)

Substituting Eq. (6.165) and Eq. (6.118) for Gf into Eq. (6.164), one obtains h[imo H  c p (T  Ts )] cp

⎡ ⎛ Dρ ⎞ ⎤ ⎟ ln(1  B) ⎥ L  ⎢⎜⎜ ⎢⎜⎝ r ⎟⎟⎠ ⎥ v ⎣ ⎦ ⎡⎛ ⎤ ⎞ ⎜ λ ⎟⎟ ⎥  ⎢⎢⎜⎜ ⎟⎟ ln(1  B) ⎥ Lv ⎜ c r ⎟ ⎢⎣⎝ p ⎠ ⎥⎦

(6.166)

where r is now the radius of the droplet. Upon cross-multiplication, Eq. (6.166) becomes hr ln(1  B)   Nu λ B

(6.167)

since ⎡ imo H  c p (T  Ts ) ⎤ ⎥ B  ⎢⎢ ⎥ Lv ⎢⎣ ⎥⎦ Since Eq. (6.118) was used for Gf, this Nusselt number [Eq. (6.167)] is for the quiescent case (Re → 0). For small B, ln(1  B)  B and the Nu  1, the classical result for heat transfer without mass addition. The term [ln(1  B)]/B has been used as an empirical correction for higher Reynolds number problems, and a classical expression for Nu with mass transfer is ⎡ ln (1  B) ⎤ ⎥ (1  0.39 Pr1/ 3 Re1r / 2 ) Nur  ⎢ ⎢⎣ ⎥⎦ B

(6.168)

371

Diffusion Flames

As Re → 0, Eq. (6.168) approaches Eq. (6.167). For the case Pr  1 and Re 1, Eq. (6.168) becomes ⎡ ln(1  B) ⎤ 1/ 2 ⎥ Rer Nur  (0.39) ⎢ ⎢⎣ ⎥⎦ B

(6.169)

The flat plate result of the preceding section could have been written in terms of a Nusselt number as well. In that case Nux 

[ f (0)]Re1x/ 2 2B

(6.170)

Thus, the burning rate expressions related to Eqs. (6.169) and (6.170) are, respectively, Gf r  0.39 Re1/2 r ln(1  B) μ 

Re1/2 r [ f (0)] 2

(6.171)

(6.172)

In convective flow a wake exists behind the droplet and droplet heat transfer in the wake may be minimal, so these equations are not likely to predict quantitative results. Note, however, that if the right-hand side of Eq. (6.171) is divided by B0.15, the expressions given by Eqs. (6.171) and (6.172) follow identical trends; thus data can be correlated as ⎛ Gf r ⎞⎟ ⎧⎪⎪ B0.15 ⎫⎪⎪ ⎜⎜ 1/2 ⎟ ⎜⎝ μ ⎟⎟⎠ ⎨⎪ ln(1  B) ⎬⎪ versus Rer ⎪⎩ ⎪⎭

(6.173)

If turbulent boundary layer conditions are achieved under certain conditions, the same type of expression should hold and Re should be raised to the 0.8 power. If, indeed, Eqs. (6.171) and (6.172) adequately predict the burning rate of a droplet in laminar convective flow, the droplet will follow a “d3/2” burning rate law for a given relative velocity between the gas and the droplet. In this case β will be a function of the relative velocity as well as B and other physical parameters of the system. This result should be compared to the “d2” law [Eq. (6.172)] for droplet burning in quiescent atmospheres. In turbulent flow, droplets will appear to follow a burning rate law in which the power of the diameter is close to 1.

372

Combustion

4. Burning Rates of Plastics: The Small B Assumption and Radiation Effects Current concern with the fire safety of polymeric (plastic) materials has prompted great interest in determining the mass burning rate of plastics. For plastics whose burning rates are measured so that no melting occurs, or for nonmelting plastics, the developments just obtained should hold. For the burning of some plastics in air or at low oxygen concentrations, the transfer number may be considered small compared to 1. For this condition, which of course would hold for any system in which B

1, ln(1  B)  B

(6.174)

and the mass burning rate expression for the case of nontransverse air movement may be written as ⎛ λ ⎞⎟ ⎜ ⎟⎟ B Gf  ⎜⎜ ⎜⎝ c p δ ⎟⎟⎠

(6.175)

Recall that for these considerations the most convenient expression for B is B

[imo H  c p (T  Ts )]

(6.176)

Lv

In most cases imo H > c p (T  Ts )

(6.177)

so B

imo H Lv

(6.178)

and ⎛ λ ⎞⎟ ⎛ im H ⎞ ⎜ ⎟⎟ ⎜⎜ o ⎟⎟ Gf  ⎜⎜ ⎜⎝ c p δ ⎟⎟⎠ ⎜⎜⎝ Lv ⎟⎟⎠

(6.179)

Equation (6.179) shows why good straight-line correlations are obtained when Gf is plotted as a function of mo for burning rate experiments in which the dynamics of the air are constant or well controlled (i.e., δ is known or constant). One should realize that Gf  mo holds only for small B.

(6.180)

373

Diffusion Flames

The consequence of this small B assumption may not be immediately apparent. One may obtain a physical interpretation by again writing the mass burning rate expression for the two assumptions made (i.e., B

1 and B  [imo H]/Lv) ⎛ λ ⎞⎟ ⎛ im H ⎞ ⎜ ⎟⎟ ⎜⎜ o ⎟⎟ Gf  ⎜⎜ ⎜⎝ c p δ ⎟⎟⎠ ⎜⎜⎝ Lv ⎟⎟⎠

(6.181)

and realizing that as an approximation (imo H )  c p (Tf  Ts )

(6.182)

where Tf is the diffusion flame temperature. Thus, the burning rate expression becomes ⎛ λ ⎞⎟ ⎡ c (T  T ) ⎤ s ⎥ ⎜ ⎟⎟ ⎢ p f Gf  ⎜⎜ ⎥ ⎜⎝ c pδ ⎟⎟⎠ ⎢⎢ L ⎥⎦ v ⎣

(6.183)

Cross-multiplying, one obtains Gf Lv 

λ(Tf  Ts ) δ

(6.184)

which says that the energy required to gasify the surface at a given rate per unit area is supplied by the heat flux established by the flame. Equation (6.184) is simply another form of Eq. (6.164). Thus, the significance of the small B assumption is that the gasification from the surface is so small that it does not alter the gaseous temperature gradient determining the heat flux to the surface. This result is different from the result obtained earlier, which stated that the stagnant film thickness is not affected by the surface gasification rate. The small B assumption goes one step further, revealing that under this condition the temperature profile in the boundary layer is not affected by the small amount of gasification. If flame radiation occurs in the mass burning process—or any other radiation is imposed, as is frequently the case in plastic flammability tests—one can obtain a convenient expression for the mass burning rate provided one assumes that only the gasifying surface, and none of the gases between the radiation source and the surface, absorbs radiation. In this case Fineman [32] showed that the stagnant film expression for the burning rate can be approximated by ⎛ λ ⎞⎟ ⎡ ⎛ ⎞⎤ ⎜ ⎟⎟ ln ⎢1  ⎜⎜ B ⎟⎟⎥ Gf  ⎜⎜ ⎜⎝ 1  E ⎟⎠⎥ ⎜⎝ c p δ ⎟⎟⎠ ⎢⎣ ⎦ and QR is the radiative heat transfer flux.

where E ≡

QR Gf Lv

374

Combustion

This simple form for the burning rate expression is possible because the equations are developed for the conditions in the gas phase and the mass burning rate arises explicitly in the boundary condition to the problem. Since the assumption is made that no radiation is absorbed by the gases, the radiation term appears only in the boundary condition to the problem. Notice that as the radiant flux QR increases, E and the term B/(1  E) increase. When E  1, the problem disintegrates because the equation was developed in the framework of a diffusion analysis. E  1 means that the solid is gasified by the radiant flux alone.

PROBLEMS 1. A laminar fuel jet issues from a tube into air and is ignited to form a flame. The height of the flame is 8 cm. With the same fuel the diameter of the jet is increased by 50% and the laminar velocity leaving the jet is decreased by 50%. What is the height of the flame after the changes are made? Suppose the experiments are repeated but that grids are inserted in the fuel tube so that all flows are turbulent. Again for the initial turbulent condition it is assumed the flame height is 8 cm. 2. An ethylene oxide monopropellant rocket motor is considered part of a ram rocket power plant in which the turbulent exhaust of the rocket reacts with induced air in an afterburner. The exit area of the rocket motor is 8 cm2. After testing it is found that the afterburner length must be reduced by 42.3%. What size must the exit port of the new rocket be to accomplish this reduction with the same afterburner combustion efficiency? The new rocket would operate at the same chamber pressure and area ratio. How many of the new rockets would be required to maintain the same level of thrust as the original power plant? 3. A spray of benzene fuel is burned in quiescent air at 1 atm and 298 K. The burning of the spray can be assumed to be characterized by single droplet burning. The (Sauter) mean diameter of the spray is 100 μm, that is, the burning time of the spray is the same as that of a single droplet of 100 μm. Calculate a mean burning time for the spray. For calculation purposes, assume whatever mean properties of the fuel, air, and product mixture are required. For some properties those of nitrogen would generally suffice. Also assume that the droplet is essentially of infinite conductivity. Report, as well, the steady-state temperature of the fuel droplet as it is being consumed. 4. Repeat the calculation of the previous problem for the initial condition that the air is at an initial temperature of 1000 K. Further, calculate the burning time for the benzene in pure oxygen at 298 K. Repeat all calculations with ethanol as the fuel. Then discuss the dependence of the results obtained on ambient conditions and fuel type.

375

Diffusion Flames

5. Two liquid sprays are evaluated in a single cylinder test engine. The first is liquid methanol, which has a transfer number B  2.9. The second is a regular diesel fuel, which has a transfer number B  3.9. The two fuels have approximately the same liquid density; however, the other physical characteristics of the diesel spray are such that its Sauter mean diameter is 1.5 times that of the methanol. Both are burning in air. Which spray requires the longer burning time and how much longer is it than the other? 6. Consider each of the condensed phase fuels listed to be a spherical particle burning with a perfect spherical flame front in air. From the properties of the fuels given, estimate the order of the fuels with respect to mass burning rate per unit area. List the fastest burning fuel first, etc.

Latent heat of vaporization Density (cal/gm) (gm/cm3) Aluminum Methanol Octane Sulfur

2570 263 87 420

2.7 0.8 0.7 2.1

Thermal conductivity (cal/s1/ cm1/K1)

Stoichiometric heat evolution in air per unit weight of fuel (cal/gm)

Heat capacity (cal/gm1/ K1)

0.48

1392

0.28

3

792

0.53

3

775

0.51

3

515

0.25

0.51  10 0.33  10 0.60  10

7. Suppose fuel droplets of various sizes are formed at one end of a combustor and move with the gas through the combustor at a velocity of 30 m/s. It is known that the 50-μm droplets completely vaporize in 5 ms. It is desired to vaporize completely each droplet of 100 μm and less before they exit the combustion chamber. What is the minimum length of the combustion chamber allowable in design to achieve this goal? 8. A radiative flux QR is imposed on a solid fuel burning in air in a stagnation film mode. The expression for the burning rate is ⎛ β ⎞⎟⎤ ⎛ Dρ ⎞⎟ ⎡ Gf  ⎜⎜ ⎟⎥ ⎟⎟ ln ⎢⎢1  ⎜⎜ ⎜⎝ 1  E ⎟⎠⎥ ⎜⎝ δ ⎠ ⎣ ⎦ where E  QR/GfLs. Develop this expression. It is a one-dimensional problem as described.

376

Combustion

9. Experimental evidence from a porous sphere burning rate measurement in a low Reynolds number laminar flow condition confirms that the mass burning rate per unit area can be represented by ⎛ Gf r ⎞⎟ ⎡ f (0, B) ⎤ ⎜⎜ ⎥ Re1/2 ⎟ ⎢ r ⎜⎝ μ ⎟⎟⎠ ⎢⎣ 2 ⎥⎦ Would a real droplet of the same fuel follow a “d2” law under the same conditions? If not, what type of power law should it follow? 10. Write what would appear to be the most important physical result in each of the following areas: (a) (b) (c) (d) (e)

laminar flame propagation laminar diffusion flames turbulent diffusion flames detonation droplet diffusion flames.

Explain the physical significance of the answers. Do not develop equations.

REFERENCES 1. Gaydon, A. G., and Wolfhard, H., “Flames.” Macmillan, New York, 1960, Chap. 6. 2. Smyth, K. C., Miller, J. H., Dorfman, R. C., Mallard, W. G., and Santoro, R. J., Combust. Flame 62, 157 (1985). 3. Kang, K. T., Hwang, J. Y., and Chung, S. H., Combust. Flame 109, 266 (1997). 4. Du, J., and Axelbaum, R. L., Combust. Flame 100, 367 (1995). 5. Hottel, H. C., and Hawthorne, W. R., Proc. Combust. Inst 3, 255 (1949). 6. Boedecker, L. R., and Dobbs, G. M., Combust. Sci. Technol 46, 301 (1986). 7. Williams, F. A., “Combustion Theory,” 2nd Ed. Benjamin-Cummins, Menlo Park, CA, 1985, Chap. 3.. 8. Lewis, B., and von Elbe, G., “Combustion, Flames and Explosion of Gases,” 2nd Ed. Academic Press, New York, 1961. 9. Burke, S. P., and Schumann, T. E. W., Ind. Eng. Chem 20, 998 (1928). 10. Roper, F. G., Combust. Flame 29, 219 (1977). 11. Roper, F. G., Smith, C., and Cunningham, A. C., Combust. Flame 29, 227 (1977). 12. Flowers, W. L., and Bowman, C. T., Proc. Combust. Inst 21, 1115 (1986). 13. Linan, A., and Williams, F. A., “Fundamental Aspects of Combustion.” Oxford University Press, Oxford, 1995. 14. Hawthorne, W. R., Weddel, D. B., and Hottel, H. C., Proc. Combust. Inst 3, 266 (1949). 15. Khudyakov, L., Chem. Abstr 46, 10844e (1955). 16. Blackshear, P. L. Jr., “An Introduction to Combustion.” Department of Mechanical Engineering, University of Minnesota, Minneapolis, 1960, Chap. 4. 17. Spalding, D. B., Proc. Combust. Inst 4, 846 (1953). 18. Spalding, D. B., “Some Fundamentals of Combustion.” Butterworth, London, 1955, Chap. 4. 19. Kanury, A. M., “Introduction to Combustion Phenomena.” Gordon & Breach, New York, 1975, Chap. 5.

Diffusion Flames

20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

377

Godsave, G. A. E., Proc. Combust. Inst 4, 818 (1953). Goldsmith, M., and Penner, S. S., Jet Propul 24, 245 (1954). Lorell, J., Wise, H., and Carr, R. E., J. Chem. Phys 25, 325 (1956). Law, C. K., Prog. Energy Combust. Sci 8, 171 (1982). Law, C. K., and Law, A. V., Combust. Sci. Technol 12, 207 (1976). Law, C. K., and Williams, F. A., Combust. Flame 19, 393 (1972). Hubbard, G. L., Denny, V. E., and Mills, A. F., Int. J. Heat Mass Transfer 18, 1003 (1975). Sirignano, W. A., and Law, C. K., Adv. Chem. Ser 166, 1 (1978). Law, C. K., Combust. Flame 26, 17 (1976). Chiu, H. H., and Liu, T. M., Combust. Sci. Technol 17, 127 (1977). Blasius, H., Z. Math. Phys 56, 1 (1956). Emmons, H. W., Z. Angew. Math. Mech 36, 60 (1956). Fineman, S., “Some Analytical Considerations of the Hybrid Rocket Combustion Problem,” M. S. E. Thesis, Department of Aeronautical Engineering, Princeton University, Princeton, NJ, 1962.

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Chapter 7

Ignition A. CONCEPTS If the concept of ignition were purely a chemical phenomenon, it would be treated more appropriately prior to the discussion of gaseous explosions (Chapter 3). However, thermal considerations are crucial to the concept of ignition. Indeed, thermal considerations play the key role in consideration of the ignition of condensed phases. The problem of storage of wet coal or large concentrations of solid materials (grain, pulverized coal, etc.) that can undergo slow exothermic decomposition in the presence of air is also one of ignition; that is, the concept of spontaneous combustion is an element of the theory of thermal ignition. Indeed, large piles of leaves and dust clouds of flour, sugar, and certain metals fall into the same category It is appropriate to reexamine the elements discussed in analysis of the explosion limits of hydrocarbons. The explosion limits shown in Fig. 3.7 of Chapter 3 exist for particular conditions of pressure and temperature. When the thermal conditions for point 1 in this figure exist, some reaction begins; thus, some heat must be evolved. The experimental configuration is assumed to be such that the heat of reaction is dissipated infinitely fast at the walls of the containing vessel to retain the temperature at the initial value T1. Then steady reaction prevails and a slight pressure rise is observed. When conditions such as those at point 2 prevail, as discussed in Chapter 3, the rate of chain carrier generation exceeds the rate of chain termination; hence the reaction rate becomes progressively greater, and subsequently an explosion—or, in the context here, ignition—occurs. Generally, pressure is used as a measure of the extent of reaction, although, of course, other measures can be used as well. The sensitivity of the measuring device determines the point at which a change in initial conditions is first detected. Essentially, this change in initial conditions (pressure) is not noted until after some time interval and, as discussed in Chapter 3, this interval can be related to the time required to reach the degenerate branching stage or some other stage in which chain branching begins to demonstrably affect the overall reaction. This time interval is considered to be an induction period and to correspond to the ignition concept. This induction period will vary considerably with temperature. Increasing the temperature increases the rates of the reactions leading to branching, thereby shortening the induction period. The isothermal events discussed in this paragraph essentially define chemical chain ignition. 379

380

Combustion

Now if one begins at conditions similar to point 1 in Fig. 3.7 of Chapter 3—except that the experimental configuration is such that the heat of reaction is not dissipated at the walls of the vessel; that is, the system is adiabatic—the reaction will self-heat until the temperature of the mixture moves the system into the explosive reaction regime. This type of event is called two-stage ignition and there are two induction periods, or ignition times, associated with it. The first is associated with the time (τc, chemical time) to build to the degenerate branching step or the critical concentration of radicals (or, for that matter, any other chain carriers), and the second (τt, thermal time) is associated with the subsequent steady reaction step and is the time it takes for the system to reach the thermal explosion (ignition) condition. Generally, τc τt. If the initial thermal condition begins in the chain explosive regime, such as point 2, the induction period τc still exists; however, there is no requirement for self-heating, so the mixture immediately explodes. In essence, τt → 0. In many practical systems, one cannot distinguish the two stages in the ignition process since τc τt; thus the time that one measures is predominantly the chemical induction period. Any errors in correlating experimental ignition data in this low-temperature regime are due to small changes in τt. Sometimes point 2 will exist in the cool-flame regime. Again, the physical conditions of the nonadiabatic experiment can be such that the heat rise due to the passage of the cool flame can raise the temperature so that the flame condition moves from a position characterized by point 1 to one characterized by point 4. This phenomenon is also called two-stage ignition. The region of point 4 is not a chain branching explosion, but a self-heating explosion. Again, an induction period τc is associated with the initial cool-flame stage and a subsequent time τt is associated with the self-heating aspect. If the reacting system is initiated under conditions similar to point 4, pure thermal explosions develop and these explosions have thermal induction or ignition times associated with them. As will be discussed in subsequent paragraphs, thermal explosion (ignition) is possible even at low temperatures, both under the nonadiabatic conditions utilized in obtaining hydrocarbon–air explosion limits and under adiabatic conditions. The concepts just discussed concern premixed fuel–oxidizer situations. In reality, these ignition types do not arise frequently in practical systems. However, one can use these concepts to gain a better understanding of many practical combustion systems, such as, for example, the ignition of liquid fuels. Many ignition experiments have been performed by injecting liquid and gaseous fuels into heated stagnant and flowing air streams [1, 2]. It is possible from such experiments to relate an ignition delay (or time) to the temperature of the air. If this temperature is reduced below a certain value, no ignition occurs even after an extended period of time. This temperature is one of interest in fire safety and is referred to as the spontaneous or autoignition temperature (AIT). Figure 7.1 shows some typical data from which the spontaneous ignition temperature is obtained. The AIT is fundamentally the temperature at which elements of the fuel–air system enter the explosion regime. Thus, the

381

Ignition

400 Sample vol. (cm2)

360

0.50 0.75 1.00 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25

Time delay before ignition (s)

320 280 240 200 160

Region of autoignition

120 80 40 0 150

160

AIT 170 180 190 200 Temperature (C)

210

220

FIGURE 7.1 Time delay before ignition of n-propyl nitrate at 1000 psig as a function of temperature (from Zabetakis [2]).

AIT must be a function of pressure as well; however, most reported data, such as those given in Appendix G are for a pressure of 1 atm. As will be shown later, a plot of the data in Fig. 7.1 in the form of log (time) versus (1/T) will give a straight line. In the experiments mentioned, in the case of liquid fuels, the fuel droplet attains a vapor pressure corresponding to the temperature of the heated air. A combustible mixture close to stoichiometric forms irrespective of the fuel. It is this mixture that enters the explosive regime, which in actuality has an induction period associated with it. Approximate measurements of this induction period can be made in a flowing system by observing the distance from point of injection of the fuel to the point of first visible radiation, then relating this distance to the time through knowledge of the stream velocity. In essence, droplet ignition is brought about by the heated flowing air stream. This type of ignition is called “forced ignition” in contrast to the “selfignition” conditions of chain and thermal explosions. The terms self-ignition, spontaneous ignition, and autoignition are used synonymously. Obviously, forced ignition may also be the result of electrical discharges (sparks), heated surfaces, shock waves, flames, etc. Forced ignition is usually considered a local initiation of a flame that will propagate; however, in certain instances, a true explosion is initiated. After examination of an analytical analysis of chain

382

Combustion

spontaneous ignition and its associated induction times, this chapter will concentrate on the concepts of self- or spontaneous ignition. Then aspects of forced ignition will be discussed. This approach will also cover the concepts of hypergolicity and pyrophoricity. Lastly, some insight into catalytic ignition is presented.

B. CHAIN SPONTANEOUS IGNITION In Chapter 3, the conditions for a chain branching explosion were developed on the basis of a steady-state analysis. It was shown that when the chain branching factor α at a given temperature and pressure was greater than some critical value αcrit, the reacting system exploded. Obviously, in that development no induction period or critical chain ignition time τc evolved. In this section, consideration is given to an analytical development of this chain explosion induction period that has its roots in the early work on chain reactions carried out by Semenov [3] and Hinshelwood [4] and reviewed by Zeldovich et al. [5]. The approach considered as a starting point is a generalized form of Eq. (3.7) of Chapter 3, but not as a steady-state expression. Thus, the overall rate of change of the concentration of chain carriers (R) is expressed by the equation d (R)/dt  ω 0  kb (R )  kt (R)  ω 0  φ (R )

(7.1)

where ω 0 is the initiation rate of a very small concentration of carriers. Such initiation rates are usually very slow. Rate expressions kb and kt are for the overall chain branching and termination steps, respectively, and φ is simply the difference kb  kt. Constants kb, kt, and, obviously, φ are dependent on the physical conditions of the system; in particular, temperature and pressure are major factors in their explicit values. However, one must realize that kb is much more temperaturedependent than kt. The rates included in kt are due to recombination (bond formation) reactions of very low activation energy that exhibit little temperature dependence, whereas most chain branching and propagating reactions can have significant values of activation energy. One can conclude, then, that φ can change sign as the temperature is raised. At low temperatures it is negative, and at high temperatures it is positive. Then at high temperatures d(R)/dt is a continuously and rapidly increasing function. At low temperatures as [d(R)/dt] → 0, (R) approaches a fixed limit ω /φ ; hence there is no runaway and no explosion. For a given pressure the temperature corresponding to φ  0 is the critical temperature below which no explosion can take place. At time zero the carrier concentration is essentially zero, and (R)  0 at t  0 serves as the initial condition for Eq. (7.1). Integrating Eq. (7.1) results in the following expression for (R): (R)  (ω 0 /φ)[exp(φt )  1]

(7.2)

383

Ignition

If as a result of the chain system the formation of every new carrier is accompanied by the formation of j molecules of final product (P), the expression for the rate of formation of the final product becomes ω  [d (P)/dt ]  jkb (R )  (jkb ω 0 /φ)[ exp(φt )  1]

(7.3)

An analogous result is obtained if the rate of formation of carriers is equal to zero (ω 0  0) and the chain system is initiated due to the presence of some initial concentration (R)0. Then for the initial condition that at t  0, (R)  (R)0, Eq. (7.2) becomes (R)  (R)0 exp(φt )

(7.4)

The derivations of Eqs. (7.1)–(7.4) are valid only at the initiation of the reaction system; kb and kt were considered constant when the equations were integrated. Even for constant temperature, kb and kt will change because the concentration of the original reactants would appear in some form in these expressions. Equations (7.2) and (7.4) are referred to as Semenov’s law, which states that in the initial period of a chain reaction the chain carrier concentration increases exponentially with time when kb kt. During the very early stages of the reaction the rate of formation of carriers begins to rise, but it can be below the limits of measurability. After a period of time, the rate becomes measurable and continues to rise until the system becomes explosive. The explosive reaction ceases only when the reactants are consumed. The time to the small measurable rate ω ms corresponds to the induction period τc. For the time close to τc, ω ms will be much larger than ω 0 and exp(φt) much greater than 1, so that Eq. (7.3) becomes ω ms  (jkb ω 0 /φ) exp(φτ )

(7.5)

The induction period then becomes τ c  (1/φ)ln(ω msφ /jkb ω 0 )

(7.6)

If one considers either the argument of the logarithm in Eq. (7.6) as a nearly constant term, or kb as much larger than kt so that φ  kb, one has τ c  const/φ

(7.7)

so that the induction time depends on the relative rates of branching and termination. The time decreases as the branching rate increases.

384

Combustion

C. THERMAL SPONTANEOUS IGNITION The theory of thermal ignition is based upon a very simple concept. When the rate of thermal energy release is greater than the rate of thermal energy dissipation (loss), an explosive condition exists. When the contra condition exists, thermal explosion is impossible. When the two rates are equal, the critical conditions for ignition (explosion) are specified. Essentially, the same type of concept holds for chain explosions. As was detailed in Section B of Chapter 3, when the rate of chain branching becomes greater than the rate of chain termination (α αcrit), an explosive condition arises, whereas α αcrit specifies steady reaction. Thus, when one considers the external effects of heat loss or chain termination, one finds a great deal of commonality between chain and thermal explosion. In consideration of external effects, it is essential to emphasize that under some conditions the thermal induction period could persist for a very long period of time, even hours. This condition arises when the vessel walls are thermally insulated. In this case, even with a very low initial temperature, the heat of the corresponding slow reaction remains in the system and gradually self-heats the reactive components until ignition (explosion) takes place. If the vessel is not insulated and heat is transferred to the external atmosphere, equilibrium is rapidly reached between the heat release and heat loss, so thermal explosion is not likely. This point will be refined in Section C.2.a. It is possible to conclude from the preceding that the study of the laws governing thermal explosions will increase understanding of the phenomena controlling the spontaneous ignition of combustible mixtures and forced ignition in general. The concepts discussed were first presented in analytical forms by Semenov [3] and later in more exact form by Frank-Kamenetskii [6]. Since the Semenov approach offers easier physical insight, it will be considered first, and then the details of the Frank-Kamenetskii approach will be presented.

1. Semenov Approach of Thermal Ignition Semenov first considered the progress of the reaction of a combustible gaseous mixture at an initial temperature T0 in a vessel whose walls were maintained at the same temperature. The amount of heat released due to chemical reaction per unit time (qr ) then can be represented in simplified overall form as qr  V ω Q  VQAc n exp(E/RT )  VQAρ n ε n exp(E/RT )

(7.8)

where V is the volume of the vessel, ω is the reaction rate, Q is the thermal energy release of the reactions, c is the overall concentration, n is the overall reaction order, A is the pre-exponential in the simple rate constant expression, and T is the temperature that exists in the gaseous mixture after reaction

385

Ignition

qr

P1

P2

q

P3

b

ql

Tb

T

c a T0 Ta

Tc

FIGURE 7.2 Rate of heat generation and heat loss of a reacting mixture in a vessel with pressure and thermal bath variations.

commences. As in Chapter 2, the concentration can be represented in terms of the total density ρ and the mass fraction ε of the reacting species. Since the interest in ignition is in the effect of the total pressure, all concentrations are . treated as equal to ρε. The overall heat loss (ql) to the walls of the vessel, and hence to the medium that maintains the walls of the vessel at T0, can be represented by the expression ql  hS (T  T0 )

(7.9)

where h is the heat transfer coefficient and S is the surface area of the walls of the containing vessel. The heat release qr is a function of pressure through the density term and ql is a less sensitive function of pressure through h, which, according to the heat transfer method by which the vessel walls are maintained at T0, can be a function of the Reynolds number. Shown in Fig. 7.2 is the relationship between q r and ql for various initial pressures, a value of the heat transfer coefficient h, and a constant wall temperature of T0. In Eq. (7.8) q r takes the usual exponential shape due to the Arrhenius kinetic rate term and ql is obviously a linear function of the mixture temperature T. The ql line intersects the q r curve for an initial pressure P3 at two points, a and b. For a system where there is simultaneous heat generation and heat loss, the overall energy conservation equation takes the form cv ρV (dT/dt )  q r  ql

(7.10)

where the term on the left-hand side is the rate of energy accumulation in the containing vessel and cv is the molar constant volume heat capacity of the gas

386

Combustion

mixture. Thus, a system whose initial temperature is T0 will rise to point a spontaneously. Since q r  ql and the mixture attains the steady, slow-reacting  Ta ), this point is an equilibrium point. If the conditions of the  i ) or ω( rate ω(T mixture are somehow perturbed so that the temperature reaches a value greater than Ta, then ql becomes greater than q r and the system moves back to the equilibrium condition represented by point a. Only if there is a very great perturbation so that the mixture temperature becomes a value greater than that represented by point b will the system self-heat to explosion. Under this condition qr ql . If the initial pressure is increased to some value P2, the heat release curve shifts to higher values, which are proportional to Pn (or ρn). The assumption is made that h is not affected by this pressure increase. The value of P2 is selected so that the ql becomes tangent to the q r curve at some point c. If the value of h is lowered, q r is everywhere greater than ql and all initial temperatures give explosive conditions. It is therefore obvious that when the ql line is tangent to the q r curve, the critical condition for mixture self-ignition exists. The point c represents an ignition temperature Ti (or Tc); and from the conditions there, Semenov showed that a relationship could be obtained between this ignition temperature and the initial temperature of the mixture—that is, the temperature of the wall (T0). Recall that the initial temperature of the mixture and the temperature at which the vessel’s wall is maintained are the same (T0). It is important to emphasize that T0 is a wall temperature that may cause a fuel–oxidizer mixture to ignite. This temperature can be hundreds of degrees greater than ambient, and T0 should not be confused with the reference temperature taken as the ambient (298 K) in Chapter 1. The conditions at c corresponding to Ti (or Tc) are qr  ql ,

(dqr /dT )  (dql /dT )

(7.11)

or VQρ n ε n A exp(E/RTi )  hS (Ti  T0 ) (dq r /dT )  (E/RTi2 )VQρ n ε n A exp(E/RTi )  (dql /dT )  hS

(7.12)

(7.13)

Since the variation in T is small, the effect of this variation on the density is ignored for simplicity’s sake. Dividing Eq. (7.12) by Eq. (7.13), one obtains ( RTi2 /E )  (Ti  T0 )

(7.14)

Equation (7.14) is rewritten as Ti 2  (E/R)Ti  (E/R)T0  0

(7.15)

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Ignition

whose solutions are Ti  (E/ 2 R)  [(E/ 2 R)2  (E/R)T0 ]1/ 2

(7.16)

The solution with the positive sign gives extremely high temperatures and does not correspond to any physically real situation. Rewriting Eq. (7.16) Ti  (E/ 2 R)  (E/ 2 R)[1  (4 RT0 /E )]1/ 2 T0

(7.17)

and expanding, one obtains Ti  (E/ 2 R)  (E/ 2 R)[1  (2RT0 /E )  2(RT0 /E )2  ] Since (RT0/E) is a small number, one may neglect the higher-order terms to obtain Ti  T0  (RT02 /E ),

(Ti  T0 )  (RT02 /E )

(7.18)

For a hydrocarbon–air mixture whose initial temperature is 700 K and whose overall activation energy is about 160 kJ/mol, the temperature rise given in Eq. (7.18) is approximately 25 K. Thus, for many cases it is possible to take Ti as equal to T0 or T0  ( RT02 /E ) with only small error in the final result. Thus, if ω (Ti )  ρ n ε n A exp (E/RTi )

(7.19)

ω (T0 )  ρ n ε n A exp(E/RT0 )

(7.20)

and

and the approximation given by Eq. (7.18) is used in Eq. (7.19), one obtains ω (Ti )  ρ n ε n A exp{E/R [T0  (RT02 /E )]}  ρ n ε n A exp{E/RT0 [1  (RT0 /E )]}  ρ n ε n A exp{(E/RT0 )[1  (RT0 /E )]}  ρ n ε n A exp[(E/RT0 )  1]  ρ n ε n A[exp(E/RT0 )]e  [ω (T0 )]e

(7.21)

That is, the rate of chemical reaction at the critical ignition condition is equal to the rate at the initial temperature times the number e. Substituting this result and the approximation given by Eq. (7.18) into Eq. (7.12), one obtains eV Qρ n ε n A exp (E/RT0 )  hSRT02 /E

(7.22)

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Combustion

Representing ρ in terms of the perfect gas law and using the logarithmic form, one obtains ln( P n T0n2 )  (E/RT0 )  ln (hSR n1 /eVQε n AE )

(7.23)

Since the overall order of most hydrocarbon oxidation reactions can be considered to be approximately 2, Eq. (7.23) takes the form of the so-called Semenov expression ln( P/T02 )  (E/ 2 RT0 )  B,

B  ln (hSR 3 /eVQε2 AE )

(7.24)

Equations (7.23) and (7.24) define the thermal explosion limits, and a plot of ln (P / T02 ) versus (1/T0) gives a straight line as is found for many gaseous hydrocarbons. A plot of P versus T0 takes the form given in Fig. 7.3 and shows the similarity of this result to the thermal explosion limit (point 3 to point 4 in Fig. 3.5) of hydrocarbons. The variation of the correlation with the chemical and physical terms in B should not he overlooked. Indeed, the explosion limits are a function of the surface area to volume ratio (S/V) of the containing vessel. Under the inherent assumption that the mass fractions of the reactants are not changing, further interesting insights can be obtained by rearranging Eq. (7.22). If the reaction proceeds at a constant rate corresponding to T0, a characteristic reaction time τr can be defined as τ r  ρ /[ρ n ε n Aexp(E/RT0 )]

(7.25)

A characteristic beat loss time τl can be obtained from the cooling rate of the gas as if it were not reacting by the expression Vcv ρ(dT/dt )  hS (T  T0 )

(7.26)

200 (CH3)2 N

P (mm of mercury)

160

C2H6 

1 2

N2

Theory Data of Rice

120 80 40 0 610

620

630

640

650

660

T0 (K) FIGURE 7.3

Critical pressure-temperature relationship for ignition of a chemical process.

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Ignition

The characteristic heat loss time is generally defined as the time it takes to cool the gas from the temperature (T  T0) to [(T  T0)/e] and is found to be τl  (V ρ cv /hS )

(7.27)

By substituting Eqs. (7.18), (7.25), and (7.27) into Eq. (7.22) and realizing that (Q/cv) can be approximated by (Tf  T0), the adiabatic explosion temperature rise, one obtains the following expression: (τ r τl )  e(Q/cv ) / ( RT02 /E )

(7.28)

Thus, if (τr / τl) is greater than the value obtained from Eq. (7.28), thermal explosion is not possible and the reaction proceeds at a steady low rate given by point a in Fig. 7.2. If (τr /τl) (eΔTf/ΔTi) and ignition still takes place, the explosion proceeds by a chain rather than a thermal mechanism. With the physical insights developed from this qualitative approach to the thermal ignition problem, it is appropriate to consider the more quantitative approach of Frank-Kamenetskii [6].

2. Frank-Kamenetskii Theory of Thermal Ignition Frank-Kamenetskii first considered the nonsteady heat conduction equation. However, since the gaseous mixture, liquid, or solid energetic fuel can undergo exothermic transformations, a chemical reaction rate term is included. This term specifies a continuously distributed source of heat throughout the containing vessel boundaries. The heat conduction equation for the vessel is then cv ρ dT/dt  div(λ grad T )  q 

(7.29)

 , which reprein which the nomenclature is apparent, except perhaps for q sents the heat release rate density. The solution of this equation would give the temperature distribution as a function of the spatial distance and the time. At the ignition condition, the character of this temperature distribution changes sharply. There should be an abrupt transition from a small steady rise to a large and rapid rise. Although computational methods of solving this equation are available, much insight into overall practical ignition phenomena can be gained by considering the two approximate methods of Frank-Kamenetskii. These two approximate methods are known as the stationary and nonstationary solutions. In the stationary theory, only the temperature distribution throughout the vessel is considered and the time variation is ignored. In the nonstationary theory, the spatial temperature variation is not taken into account, a mean temperature value throughout the vessel is used, and the variation of the mean temperature with time is examined. The nonstationary problem is the same as that posed by Semenov; the only difference is in the mathematical treatment.

390

Combustion

a. The Stationary Solution—The Critical Mass and Spontaneous Ignition Problems The stationary theory deals with time-independent equations of heat conduction with distributed sources of heat. Its solution gives the stationary temperature distribution in the reacting mixture. The initial conditions under which such a stationary distribution becomes impossible are the critical conditions for ignition. Under this steady assumption, Eq. (7.29) becomes div(λ grad T )  q

(7.30)

and, if the temperature dependence of the thermal conductivity is neglected, λ∇2T  q

(7.31)

 . Defined as the amount of It is important to consider the definition of q  is the heat evolved by chemical reaction in a unit volume per unit time, q product of the terms involving the energy content of the fuel and its rate of reaction. The rate of the reaction can be written as ZeE/RT. Recall that Z in this example is different from the normal Arrhenius pre-exponential term in that it contains the concentration terms and therefore can be dependent on the mixture composition and the pressure. Thus, q  QZeE/RT

(7.32)

where Q is the volumetric energy release of the combustible mixture. It follows then that ∇2T  (Q/λ )ZeE RT

(7.33)

and the problem resolves itself to first reducing this equation under the boundary condition that T  T0 at the wall of the vessel. Since the stationary temperature distribution below the explosion limit is sought, in which case the temperature rise throughout the vessel must be small, it is best to introduce a new variable ν  T  T0 where ν

T0. Under this condition, it is possible to describe the cumbersome exponential term as ⎡ E ⎤ ⎪⎧ ⎛ E ⎞⎟ ⎥  exp ⎪⎨ E exp ⎜⎜ ⎟⎟  exp ⎢⎢ ⎥ ⎜⎝ RT ⎠ ⎪⎪ RT0 ⎣ R(T0  ν ) ⎦ ⎩

⎡ ⎤ ⎪⎫⎪ 1 ⎢ ⎥ ⎢ 1  (ν /Τ ) ⎥ ⎬⎪ 0 ⎦⎪ ⎣ ⎭

If the term in brackets is expanded and the higher-order terms are eliminated, this expression simplifies to ⎡ ⎛ E ⎞⎟ E exp ⎜⎜ ⎟⎟ ≅ exp ⎢⎢ ⎜⎝ RT ⎠ ⎢⎣ RT0

⎡ ⎤ ⎡ ⎤ ⎛ ⎞⎤ ⎜⎜1  ν ⎟⎟⎥  exp ⎢ E ⎥ exp ⎢ E ν ⎥ ⎟ ⎢ RT ⎥ ⎢ RT 2 ⎥ ⎜⎜⎝ T0 ⎟⎠⎥⎥⎦ 0 ⎦ ⎣ ⎣ 0 ⎦

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Ignition

and (Eq. 7.33) becomes ∇2 ν  

⎡ ⎡ E ⎤ Q E ⎤⎥ ⎢ ⎥ Z exp ⎢⎢ exp ν ⎥ ⎢ RT 2 ⎥ λ ⎣ RT0 ⎦ ⎣ 0 ⎦

(7.34)

In order to solve Eq. (7.34), new variables are defined θ  (E/RT02 )ν ,

η x  x/ r

where r is the radius of the vessel and x is the distance from the center. Equation (7.34) then becomes ∇2η θ  {(Q/λ )( E/RT02 )r 2 ZeE/RT0 }eθ

(7.35)

and the boundary conditions are η  1, θ  0, and η  0, dθ/dη  0. Both Eq. (7.35) and the boundary conditions contain only one nondimensional parameter δ: δ  (Q/λ )(E/RT02 )r 2 ZeE/RT0

(7.36)

The solution of Eq. (7.35), which represents the stationary temperature distribution, should be of the form θ  f (η, δ) with one parameter, that is, δ. The condition under which such a stationary temperature distribution ceases to be possible, that is, the critical condition of ignition, is of the form δ  const  δcrit. The critical value depends upon T0, the geometry (if the vessel is nonspherical), and the pressure through Z. Numerical integration of Eq. (7.35) for various δ’s determines the critical δ. For a spherical vessel, δcrit  3.32; for an infinite cylindrical vessel, δcrit  2.00; and for infinite parallel plates, δcrit  0.88, where r becomes the distance between the plates. As in the discussion of flame propagation, the stoichiometry and pressure dependence are in Z and Z  Pn, where n is the order of the reaction. Equation (7.36) expressed in terms of δcrit permits the relationship between the critical parameters to be determined. Taking logarithms, ln rP n ∼ (E/RT0 ) If the reacting medium is a solid or liquid undergoing exothermic decomposition, the pressure term is omitted and ln r ∼ (E/RT0 ) These results define the conditions for the critical size of storage for compounds such as ammonium nitrate as a function of the ambient temperature T0 [7]. Similarly, it represents the variation in mass of combustible material that will

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Combustion

spontaneously ignite as a function of the ambient temperature T0. The higher the ambient temperature, the smaller the critical mass has to be to prevent disaster. Conversely, the more reactive the material, the smaller the size that will undergo spontaneous combustion. Indeed this concept is of great importance from a fire safety point of view due to the use of linseed oil and tung oil as a polishing and preserving agent of fine wood furniture. These oils are natural products that have never been duplicated artificially [7a]. Their chemical structures are such that when exposed to air an oxidation reaction forms a transparent oxide coating (of the order of 24 Å) that protects wood surfaces. There is a very small finite amount of heat released during this process, which the larger mass of the applied object readily absorbs. However, if the cloths that are used to apply these oils are not disposed of properly, they will self ignite. Disposing of these clothes in a waste receptacle is dangerous unless the receptacle contains large amounts of water for immersion of the cloths. The protective oxide coat formed during polishing is similar to the protective oxide coat on aluminum. Also, a large pile of damp leaves or pulverized coal, which cannot disperse the rising heat inside the pile, will ignite as well. Generally in these cases the use of the term spontaneous ignition could be misleading in that the pile of cloths with linseed oil, a pile of leaves or a pile of pulverized coal will take a great deal of time before the internal elements reach a high enough temperature that combustion starts and there is rapid energy release leading to visible flames.

b. The Nonstationary Solution The nonstationary theory deals with the thermal balance of the whole reaction vessel and assumes the temperature to be the same at all points. This assumption is, of course, incorrect in the conduction range where the temperature gradient is by no means localized at the wall. It is, however, equivalent to a replacement of the mean values of all temperature-dependent magnitudes by their values at a mean temperature, and involves relatively minor error. If the volume of the vessel is designated by V and its surface area by S, and if a heat transfer coefficient h is defined, the amount of heat evolved over the whole volume per unit time by the chemical reaction is V QZeE/RT

(7.37)

and the amount of heat carried away from the wall is hS (T  T0 )

(7.38)

Thus, the problem is now essentially nonadiabatic. The difference between the two heat terms is the heat that causes the temperature within the vessel to rise a certain amount per unit time, cv ρ V (dT/dt )

(7.39)

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Ignition

These terms can be expressed as an equality, cv ρ V (dT/dt )  V QZeE/RT  hS(T  T0 )

(7.40)

dT/dt  (Q/cv ρ)ZeE/RT  (hS/cv ρV )(T  T0 )

(7.41)

or

Equations (7.40) and (7.41) are forms of Eq. (7.29) with a heat loss term. Nondimensionalizing the temperature and linearizing the exponent in the same manner as in the previous section, one obtains d θ /dt  (Q/cv ρ )(E/RT02 )ZeE/RT0 eθ  (hS/cv ρV )θ

(7.42)

with the initial condition θ  0 at t  0. The equation is not in dimensionless form. Each term has the dimension of reciprocal time. In order to make the equation completely dimensionless, it is necessary to introduce a time parameter. Equation (7.42) contains two such time parameters: τ1  [(Q/cv ρ )(E/RT02 )ZeE/RT0 ]1 ,

τ 2  (hS/cv ρV )1

Consequently, the solution of Eq. (7.42) should be in the form θ  f (t/τ1, 2 , τ 2 /τ1 ) where τl,2 implies either τ1 or τ2. Thus, the dependence of dimensionless temperature θ on dimensionless time t/τ1,2 contains one dimensionless parameter τ2 / τ1. Consequently, a sharp rise in temperature can occur for a critical value τ2 / τ1. It is best to examine Eq. (7.42) written in terms of these parameters, that is, dθ/dt  (eθ /τ1 )  (θ /τ 2 )

(7.43)

In the ignition range the rate of energy release is much greater than the rate of heat loss; that is, the first term on the right-hand side of Eq. (7.43) is much greater than the second. Under these conditions, removal of heat is disregarded and the thermal explosion is viewed as essentially adiabatic. Then for an adiabatic thermal explosion, the time dependence of the temperature should be in the form θ  f (t/τ1 )

(7.44)

Under these conditions, the time within which a given value of θ is attained is proportional to the magnitude τ1. Consequently, the induction period in the instance of adiabatic explosion is proportional to τ1. The proportionality

394

Combustion

constant has been shown to be unity. Conceptually, this induction period can be related to the time period for the ignition of droplets for different air (or ambient) temperatures. Thus τ can be the adiabatic induction time and is simply τ

cv ρ RT02 1 E/RT 0 e Q E Z

(7.45)

Again, the expression can be related to the critical conditions of time, pressure, and ambient temperature T0 by taking logarithms: ln (τ P n1 ) ∼ (E/RT0 )

(7.46)

The pressure dependence, as before, is derived not only from the perfect gas law for ρ, but from the density–pressure relationship in Z as well. Also, the effect of the stoichiometry of a reacting gas mixture would be in Z. But the mole fraction terms would be in the logarithm, and therefore have only a mild effect on the induction time. For hydrocarbon–air mixtures, the overall order is approximately 2, so Eq. (7.46) becomes ln (τ P ) ∼ (E/RT0 )

(7.47)

It is interesting to note that Eq. (7.47) is essentially the condition used in bluff-body stabilization conditions in Chapter 4, Section F. This result gives the intuitively expected answer that the higher the ambient temperature, the shorter is the ignition time. Hydrocarbon droplet and gas fuel injection ignition data correlate well with the dependences as shown in Eq. (7.47) [8, 9]. In a less elegant fashion, Todes [10] (see Jost [11]) obtained the same expression as Eq. (7.45). As Semenov [3] has shown by use of Eq. (7.25), Eq. (7.45) can be written as τi  τ r (cv RT02 /QE )

(7.48)

Since (E/RT0)1 is a small quantity not exceeding 0.05 for most cases of interest and (cvT0/Q) is also a small quantity of the order 0.1, the quantity (cv RT02 /QE ) may be considered to have a range from 0.01 to 0.001. Thus, the thermal ignition time for a given initial temperature T0 is from a hundredth to a thousandth of the reaction time evaluated at T0. Since from Eq. (7.28) and its subsequent discussion, τ r  [(Q/cv ) / ( E/RT02 )]eτl

(7.49)

τ i  eτ l

(7.50)

then

Ignition

395

which signifies that the induction period is of the same order of magnitude as the thermal relaxation time. Since it takes only a very small fraction of the reaction time to reach the end of the induction period, at the moment of the sudden rapid rise in temperature (i.e., when explosion begins), not more than 1% of the initial mixture has reacted. This result justifies the inherent approximation developed that the reaction rate remains constant until explosion occurs. Also justified is the earlier assumption that the original mixture concentration remains the same from T0 to Ti. This observation is important in that it reveals that no significantly different results would be obtained if the more complex approach using both variations in temperature and concentration were used.

D. FORCED IGNITION Unlike the concept of spontaneous ignition, which is associated with a large condensed-phase mass of reactive material, the concept of forced ignition is essentially associated with gaseous materials. The energy input into a condensed-phase reactive mass may be such that the material vaporizes and then ignites, but the phenomena that lead to ignition are those of the gas-phase reactions. There are many means to force ignition of a reactive material or mixture, but the most commonly studied concepts are those associated with various processes that take place in the spark ignition, automotive engine. The spark is the first and most common form of forced ignition. In the automotive cylinder it initiates a flame that travels across the cylinder. The spark is fired before the piston reaches top dead center and, as the flame travels, the combustible mixture ahead of this flame is being compressed. Under certain circumstances the mixture ahead of the flame explodes, in which case the phenomenon of knock is said to occur. The gases ahead of the flame are usually ignited as the temperature rises due to the compression or some hot spot on the metallic surfaces in the cylinder. As discussed in Chapter 2, knock is most likely an explosion, but not a detonation. The physical configuration would not permit the transformation from a deflagration to a detonation. Nevertheless, knock, or premature forced ignition, can occur when a fuel–air mixture is compressively heated or when a hot spot exists. Consequently, it is not surprising that the ignitability of a gaseous fuel–air mixture—or, for that matter, any exoergic system—has been studied experimentally by means of approaching adiabatic compression to high temperature and pressure, by shock waves (which also raise the material to a high temperature and pressure), or by propelling hot metallic spheres or incandescent particles into a cold reactive mixture. Forced ignition can also be brought about by pilot flames or by flowing hot gases, which act as a jet into the cold mixture to be ignited. Or it may be engendered by creating a boundary layer flow parallel to the cold mixture, which may also be flowing. Indeed, there are several other possibilities that one might evoke. For consideration of these systems, the reader is referred to Ref. [12].

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Combustion

It is apparent, then, that an ignition source can lead either to a pure explosion or to a flame (deflagration) that propagates. The geometric configuration in which the flame has been initiated can be conducive to the transformation of the flame into a detonation. There are many elements of concern with respect to fire and industrial safety in these considerations. Thus, a concept of a minimum ignition energy has been introduced as a test method for evaluating the ignitability of various fuel–air mixtures or any system that has exoergic characteristics. Ignition by near adiabatic compression or shock wave techniques creates explosions that are most likely chain carrier, rather than thermal, initiated. This aspect of the subject will be treated at the end of this chapter. The main concentration in this section will be on ignition by sparks based on a thermal approach by Zeldovich [13]. This approach, which gives insights not only into the parameters that give spark ignition, but also into forced ignition systems that lead to flames, has applicability to the minimum ignition energy.

1. Spark Ignition and Minimum Ignition Energy The most commonly used spark systems for mobile power plants are capacitance sparks, which are developed from discharged condensers. The duration of these discharges can be as short as 0.01 μs or as long as 100 μs for larger engines. Research techniques generally employ two circular electrodes with flanges at the tips. The flanges have a parallel orientation and a separation distance greater than the quenching distance for the mixture to be ignited. (Reference [12] reports extensive details about spark and all other types of forced ignition.) The energy in a capacitance spark is given by E

1 2

cf (v22  v12 )

(7.51)

where E is the electrical energy obtained in joules, cf is the capacitance of the condenser in farads, v2 is the voltage on the condenser just before the spark occurs, and v1 is the voltage at the instant the spark ceases. In the Zeldovich method of spark ignition, the spark is replaced by a point heat source, which releases a quantity of heat. The time-dependent distribution of this heat is obtained from the energy equation (∂T/∂t )  α∇2T

(7.52)

When this equation is transformed into spherical coordinates, its boundary conditions become r  ∞, T  T0

and

r  0, (∂T/∂r )  0

The distribution of the input energy at any time must obey the equality Qv′  4π c p ρ ∫

∞

0

(T  T0 )r 2 dr

(7.53)

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Ignition

The solution of Eq. (7.52) then becomes (T  T0 )  {Qv′ /[c p ρ(4παt )]3 / 2 }exp(r 2 / 4αt )

(7.54)

The maximum temperature (TM) must occur at r → 0, so that (TM  T0 )  {Qv′ /[c p ρ(4παt )]3 / 2

(7.55)

Considering that the gaseous system to be ignited exists everywhere from r  0 to r  , the condition for ignition is specified when the cooling time (τc) associated with TM is greater than the reaction duration time τr in the combustion zone of a laminar flame. This characteristic cooling time is the period in which the temperature at r  0 changes by the value θ. This small temperature difference θ is taken as (RTM2 /E ); that is, θ  RTM2 /E

(7.56)

This expression results from the same type of analysis that led to Eq. (7.18). A plot of TM versus τ [Eq. (7.55)] is shown in Fig. 7.4. From this figure the characteristic cooling time can be taken to a close approximation as τ c  θ / dTM /dt |TM Tr

(7.57)

The slope is taken at a time when the temperature at r  0 is close to the adiabatic flame temperature of the mixture to be ignited. By differentiating Eq. (7.55), the denominator of Eq. (7.57) can be evaluated to give τ c  [θ /6απ(Tf  T0 )]{Qv′ /[c p ρ(Tf  T0 )]}2/3

(7.58)

TM

Tr Tr  θ Δτ τc τc

τ

FIGURE 7.4 Variation of the maximum temperature with time for energy input into a spherical volume of a fuel-air mixture.

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Combustion

where Qv′ is now given a specific definition as the amount of external input energy required to heat a spherical volume of radius rf uniformly from T0 to Tf; that is, Qv  ( 43 )πrf3 c p ρ(Tf  T0 )

(7.59)

Thus, Eq. (7.58) becomes τ c  0.14[θ /(Tf  T0 )](rf2 /α )

(7.60)

Considering that the temperature difference θ must be equivalent to (Tf  Ti) in the Zeldovich–Frank-Kamenetskii–Semenov thermal flame theory, the reaction time corresponding to the reaction zone δ in the flame can also be approximated by τ r ≅ [2θ /(Tf  T0 )](α /SL2 )

(7.61)

where (α /SL2 ) is the characteristic time associated with the flame and a  (α /SL )

(7.62)

specifies the thermal width of the flame. Combining Eqs. (7.60), (7.61), and (7.62) for the condition τc  τr yields the condition for ignition as rf  3.7a

(7.63)

Physically, Eq. (7.63) specifies that for a spark to lead to ignition of an exoergic system, the corresponding equivalent heat input radius must be several times the characteristic width of the laminar flame zone. Under this condition, the nearby layers of the initially ignited combustible material will further ignite before the volume heated by the spark cools. The preceding developments are for an idealized spark ignition system. In actual systems, much of the electrical energy is expended in radiative losses, shock wave formation, and convective and conductive heat losses to the electrodes and flanges. Zeldovich [13] has reported for mixtures that the efficiency ηs  (Qv′ /E )

(7.64)

can vary from 2% to 16%. Furthermore, the development was idealized by assuming consistency of the thermophysical properties and the specific heat. Nevertheless, experimental results taking all these factors into account [13, 14] reveal relationships very close to rf  3a

(7.65)

399

Ignition

The further importance of Eqs. (7.63) and (7.65) is in the determination of the important parameters that govern the minimum ignition energy. By substituting Eqs. (7.62) and (7.63) into Eq. (7.59), one obtains the proportionality Qv′,min ∼ (α /SL )3 c p ρ(Tf  T0 )

(7.66)

Considering α  (λ/ρcp) and applying the perfect gas law, the dependence ′ of Qv,min on P and T is found to be Qv′,min ∼ [λ3T02 (Tf  T0 )]/SL3 P 2 c 2p

(7.67)

The minimum ignition energy is also a function of the electrode spacing. It becomes asymptotic to a very small spacing below which no ignition is possible. This spacing is the quenching distance discussed in Chapter 4. The minimum ignition energy decreases as the electrode spacing is increased, reaches its lowest value at some spacing and then begins to rise again. At small spacings the electrode removes large amounts of heat from the incipient flame, and thus a large minimum ignition energy is required. As the spacing increases, the surface area to volume ratio decreases, and, consequently, the ignition energy required decreases. Most experimental investigations [12, 15] report the minimum ignition energy for the electrode spacing that gives the lowest value. An interesting experimental observation is that there appears to be an almost direct relation between the minimum ignition energy and the quenching distance [15, 16]. Calcote et al. [15] have reported significant data in this regard, and their results are shown in Fig. 7.5. These data are for stoichiometric mixtures with air at 1 atm. The general variation of minimum ignition energy with pressure and temperature would be that given in Eq. (7.67), in which one must recall that SL is also a function of the pressure and the Tf of the mixture. Figure 7.6 from Blanc et al. [14] shows the variation of Qv′ as a function of the equivalence ratio. The variation is very similar to the variation of quenching distance with the equivalence ratio φ [11] and is, to a degree, the inverse of SL versus φ. However, the increase of Qv′ from its lowest value for a given φ is much steeper than the decay of SL from its maximum value. The rapid increase in Qv′ must be due to the fact that SL is a cubed term in the denominator of Eq. (7.67). Furthermore, the lowest Qv′ is always found on the fuel-rich side of stoichiometric, except for methane [13, 15]. This trend is apparently attributable to the difference in the mass diffusivities of the fuel and oxygen. Notice the position of the methane curve with respect to the other hydrocarbons in Fig. 7.6. Methane is the only hydrocarbon shown whose molecular weight is appreciably lower than that of the oxygen in air.

400

Combustion

100 80 60 40

Minimum spark ignition energy (104 J)

20 Cyclohexane

10 8 6

n-Heptane Heptyne-1 Pentene-2 Methyl formate

Vinyl acetate Cyclohexene Isobutane n-Pentane

Benzene

4

Propionaldehyde Propane

Diethyl ether

2

Butadiene-1,3 Methyl acetylene

1.0 0.8

Iso-octane -Di-Isopropylether -Di-Isobutylene

Propylene oxide

Ethylene oxide

0.6 0.4

0.2

Hydrogen

Acetylene Carbon disulfide

0.1 0.2

0.4 0.6 0.8 1.0

2

4

6

8 10

20

Quenching distance (mm) FIGURE 7.5 Correlation of the minimum spark ignition energy with quenching diameter (from Calcote et al. [15]).

Many [12,15] have tried to determine the effect of molecular structure on Qv′ . Generally the primary effect of molecular structure is seen in its effect on Tf (in SL) and α. Appendix H lists minimum ignition energies of many fuels for the stoichiometric condition at a pressure of 1 atm. The Blanc data in this appendix are taken from Fig. 7.6. It is remarkable that the minima of the energy curves for the various compounds occur at nearly identical values. In many practical applications, sparks are used to ignite flowing combustible mixtures. Increasing the flow velocity past the electrodes increases the energy required for ignition. The flow blows the spark downstream, lengthens the spark path, and causes the energy input to be distributed over a much larger volume [12]. Thus the minimum energy in a flow system is greater than that under a stagnant condition. From a safety point of view, one is also interested in grain elevator and coal dust explosions. Such explosions are not analyzed in this text, and the reader

401

Ignition

4 3

Propane Methane

Ethane

Minimum energy (mJ)

2

Butane Hexane

Heptane

1 0.8 0.6 0.5 0.4 0.3 0.2

1 At

0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

4 3 Benzene

2 Butane Propane

Cyclohexane Hexane

1 0.8 0.6 0.5 0.4 0.3 0.2 Cyclopropane

Diethyl ether

1 At

0.1 0

0.5 1.0 1.5 2.0 0 0.5 1.0 1.5 2.0 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Fraction of stoichiometric percentage of combustible in air FIGURE 7.6 Minimum ignition energy of fuel–air mixtures as a function of stoichiometry (from Blanc et al. [14]).

is referred to the literature [17]. However, many of the thermal concepts discussed for homogeneous gas-phase ignition will be fruitful in understanding the phenomena that control dust ignition and explosions.

2. Ignition by Adiabatic Compression and Shock Waves Ignition by sparks occurs in a very local region and spreads by flame characteristics throughout the combustible system. If an exoergic system at standard conditions is adiabatically compressed to a higher pressure and hence to a higher temperature, the gas-phase system will explode. There is little likelihood that a flame will propagate in this situation. Similarly, a shock wave can propagate through the same type of mixture, rapidly compressing and heating the mixture to an explosive condition. As discussed in Chapter 5, a detonation will develop under such conditions only if the test section is sufficiently long. Ignition by compression is similar to the conditions that generate knock in a spark-ignited automotive engine. Thus it would indeed appear that compression ignition and knock are chain-initiated explosions. Many have established

402

Combustion

TABLE 7.1 Compression Versus Shock-Induced Temperaturea Shock and adiabatic compression ratio

Shock wave velocity (m/s)

T(K) Behind shock

T(K) After compression

2

452

336

330

5

698

482

426

10

978

705

515

50

2149

2261

794

100

3021

3861

950

1000

9205

19,089

1711

2000

12,893

25,977

2072

a

Initial temperature, 273 K.

the onset of ignition with a rapid temperature rise over and above that expected due to compression. Others have used the onset of some visible radiation or, in the case of shock tubes, a certain limit concentration of hydroxyl radical formation identified by spectroscopic absorption techniques. The observations and measurement techniques are interrelated. Ignition occurs in such systems in the 1000 K temperature range. However, it must be realized that in hydrocarbon–air systems the rise in temperature due to exothermic energy release of the reacting mixture occurs most sharply when the carbon monoxide that eventually forms is converted to carbon dioxide. This step is the most exothermic of all the conversion steps of the fuel–air mixture to products [18]. Indeed, the early steps of the process are overall isoergic owing to the simultaneous oxidative pyrolysis of the fuel, which is endothermic, and the conversion of some of the hydrogen formed to water, which is an exothermic process [18]. Shock waves are an ideal way of obtaining induction periods for hightemperature—high-pressure conditions. Since a shock system is nonisentropic, a system at some initial temperature and pressure condition brought to a final pressure by the shock wave will have a higher temperature than a system in which the same mixture at the same initial conditions is brought by adiabatic compression to the same pressure. Table 7.1 compares the final temperatures for the same ratios of shock and adiabatic compression for air.

E. OTHER IGNITION CONCEPTS The examples that appeared in Section C were with regard to linseed and tung oils, damp leaves, and pulverized coal. In each case a surface reaction occurred. To be noted is the fact that the analyses that set the parameters for determining the ignition condition do not contain a time scale. In essence then, the concept

Ignition

403

of spontaneity should not be considered in the same context as rapidity. Dictionaries [19] define spontaneous combustion as “the ignition of a substance or body from the rapid oxidation of its constituents, without heat from any external source”. This definition would be ideal if the word “rapid” were removed. Oil spills on an ocean or oil on a beach also react with oxygen in air. Considering there is always moisture in the air, it is not surprising the coating one observes on fuels when one steps on an oil globule on the beach has been found to be an organic hydrocarbon peroxide (R¶OOH, Chapter 3, Section E). It is obvious during the oxidative process the mass of fuel concerned in the cases can readily absorb the heat released in the peroxide formation so that any thermal rise is not sufficient to cause rapid reaction. One can then realize from the considerations in Sections B, C, and D that for different substances, their configurations, ambient conditions, etc., can affect what causes an ignition process to take hold. These factors are considered extensively in the recently published Ignition Handbook [20], which also contains many references.

1. Hypergolicity and Pyrophoricity There are practical cases in which instantaneous ignition must occur or there would be a failure of the experimental objective. The best example would be the necessity to instantaneously ignite the injection of liquid oxygen and liquid hydrogen in a large booster rocket. Instantaneous in this case means in a time scale that a given amount of propellants that can accumulate and still be ignited would not destroy the rocket due to a pressure rise much greater than the pressure for which the chamber was designed. This, called delayed ignition, was the nemesis of early rocket research. One approach to the liquid H2–liquid O2 case has been to inject triethyl aluminum (TEA), a liquid, with the injection of the fuels. Because of the electronic structure of the aluminum atom, TEA instantaneously reacts with the oxygen. In general, those elements whose electronic structure show open d-orbitals [21, 22] in their outer electron ring show an affinity for reaction, particularly with oxygen. In the rocket field TEA is referred to as a hypergolic propellant. Many of the storable (non-cryogenic) propellants are hypergolic, particularly when red fuming nitric acid that is saturated with nitrogen tetroxide (N2O4) or unsymmetrical dimethyl hydrazine (UDMH). The acids attach to any weakly bound hydrogen atoms in the fuels, almost like an acid–base reaction and initiate overall combustion. In the case of supersonic combustion ramjet devices, instantaneous ignition must occur because the flow time in the constant area duct that comprises the ramjet chamber is short. As noted in Chapter 1, supersonic combustion simply refers to the flow condition and not to any difference in the chemical reaction mechanism from that in subsonic ramjet devices. What is unusual in supersonic combustion because of the typical flow condition is that the normal ignition time is usually longer than the reaction time. To assure rapid ignition in this case, many have proposed the injection of silane (SiH4), which is

404

Combustion

hypergolic with the oxygen in air. Since exposure of silane in a container to air causes an instant flame process, many refer to silane as pyrophoric. It is interesting to note that dictionaries define pyrophoric as “capable of igniting spontaneously in air” [19]. Notice the use of “spontaneously”. The term pyrophoric has usually been applied to the ignition of very fine sizes of metal particles. Except for the noble metals, most metals when refined and exposed to air form an oxide coat. Generally this coating thickness is of the order of 25 Å. If the oxide coat formed is of greater size than that of the pure metal consumed, then the coat scales and the nascent metal is prone to continuously oxidize. Iron is a case in point and is the reason pure iron rusts. The ratio of the oxide formed to the metal consumed is called the Piling and Bedworth number. When the number is over 1, the metal rusts. Aluminum and magnesium are the best examples of metals that do not rust because a protective oxide coat forms; that is, they have a Piling to Bedworth number of 1. Scratch an aluminum ladder and notice a bright fissure forms and quickly self-coats. The heat release in the sealing aluminum oxide is dissipated to the ladder structure. Aluminum particles in solid propellant rockets do not burn in the same manner as do aluminum wires that carry electric currents. The question arises at what temperature do these materials burst into flames. First, the protective oxide coat must be destroyed and second, the temperature must be sufficient to vaporize the exposed surface of the nascent metal. The metal vapor that reacts with the oxidizer present must reach a temperature that will retain the vaporization of the metal. As will be discussed more extensively in Chapter 9, it is known that the temperature created by the formation of the metal–oxidizer reaction has the unique property, if the oxidizer is pure oxygen, to be equal to the boiling point (really the vaporization point) of the metal oxide at the process pressure. The vaporization temperature of the oxide then must be greater than the vaporization temperature of the metal. Those metal particles that meet this criterion burn very much like a liquid hydrocarbon droplet. Some metals do not meet this criterion and their transformation into an oxide is vastly different from those that meet the criterion, which many refer to as Glassman’s Criterion for Vapor Phase Combustion of Metals. The temperature and conditions that lead to the ignition of those metals that burn in the vapor phase are given in the Ignition Handbook [20] where again numerous references can be found. It is with the understanding of the above that one can give some insight to what establishes the pyrophoricity of small metal particles. The term pyrophoricity should pertain to the instantaneous combustibility of fine metal particles that have no oxide coat. This coating prevention is achieved by keeping the particles formed and stored in an inert atmosphere such as argon. Nitrogen is not used because nitrides can be formed. When exposed to air, the fine metal particle cloud instantaneously bursts into a flame. Thus it has been proposed

Ignition

405

[23] that a metal be considered pyrophoric when in its nascent state (no oxide coat) it is small enough that the initial oxide coat that forms due to heterogeneous reaction with air under ambient conditions generates sufficient heat to vaporize the remaining metal. Metal vapors thus exposed are extremely reactive with oxidizing media and are consumed rapidly. Although pyrophoric metals can come in various shapes (spherical, porous spheres or flakes), the calculation to be shown will be based on spherical particles. Since it is the surface area to volume ratio that determines the critical condition, then it would be obvious for a metal flake (which would be pyrophoric) to have a smaller mass than a sphere of the same metal. Due to surface temperature, however, pyrophoric flakes will become spheres as the metal melts. Stated physically, the critical condition for pyrophoricity under the proposed assumptions is that the heat release of the oxide coat formed on a nascent sphere at the ambient temperature must be sufficient to heat the metal to its vaporization point and supply enough heat to vaporize the remaining metal. In such an approach one must take into account the energy necessary to raise the metal from the ambient temperature to the vaporization temperature. If r is assumed to be the radius of the metal particle and δ the thickness of the oxide coat [(r – δ) is the pure metal radius], then the critical heat balance for pyrophoricity contains three terms: (a) (4/3)π[r 3  (r  δ)3]ρox(ΔH°298)ox  Heat available (b) (4/3)π(r  δ)3ρm{(H°bpt  H°298)m  Lv}  Heat needed to vaporize the metal (c) (4/3)π[r 3  (r  δ)3]ρox(H°bpt  H°298)ox  Heat needed to heat the oxide coat where (ΔH°298)ox is the standard state heat of formation of the oxide at 298 K, Ho is the standard state enthalpy at temperature T, “bpt” specifies the metal vaporization temperature, and the subscripts “m” and “ox” refer to the metal and oxide respectively. At the critical condition the term (a) must be equal to the sum of the terms (b) and (c). This equality can be rearranged and simplified to give the form ρox {(ΔH°298 )ox  (H ºbpt  H º298 ) ox}/{ρm ((H ºbpt  H298 º ) m  L v )}  (1  δ/r )3 /{1  (1  δ/r )3} Considering the right-hand side of this equation as a simple mathematical function, it can be plotted versus (δ/r). The left-hand side is known for a given metal; it contains known thermochemical and thermophysical properties, thus (δ/r) is determined. The mass of the oxide formed is greater than the mass of the metal consumed, consequently the original size of the metal that would be pyrophoric (rm) can be calculated from δ, r and the physical properties of the oxide and metal and their molecular weights. These results have been presented in the

406

Combustion

Ca Mg

(1-δ/r)3 / {13-(1-δ/r)3}

4 Rb

Cs

K

3

Na Li

2

1

Al

0 0.0

Ce Zn Th Pu Zr Hf Mn Sn Cr Ti Pb Si U

0.2

Ta Fe Co Mo Ni Cu

0.4

0.6

0.8

{ρ(-ΔH298-(Hv-H298))}ox/{ρ((Hv-H298)Lv)}m

5

1.0

δ/r

FIGURE 7.7 Values of /r for several metals; where  is the oxide layer thickness and r is the radius of a pyrophoric particle, including .).

form as given in Fig. 7.7 [23]. The important result is simply that the smaller the value of (δ/r), the greater the pyrophoric tendency of the metal. The general size of those particles that are pyrophoric is of the order 0.01 μm [23]. It is further interesting to note that all metals that have values of (δ/r) less than 0.2 meet Glassman’s criterion for vapor-phase combustion of metals regardless of size (see Chapter 9). Indeed, Fig 7.7 gives great insight into metal combustion and the type of metal dust explosions that have occurred. One can further conclude that only those metals that have a (δ/r) value less than 0.2 are the ones prone to dust type explosions.

2. Catalytic Ignition Consideration with respect to hydrazine ignition forms the basis of an approach to some understanding of catalytic ignition. Although hydrazine and its derivative UDMH are normally the fuels in storable liquid propellant rockets because they are hypergolic with nitric oxides as discussed in the last section, hydrazine is also frequently used as a monopropellant. To retain the simplicity of a monopropellant rocket to use for control purposes or a backpack lift for an astronaut, the ability to catalytically ignite the hydrazine becomes a necessity. Thus small hydrazine monopropellant rockets contain in their chamber large surface area configurations that are plated with platinum. Injection of hydrazine in such a rocket chamber immediately initiates hydrazine decomposition and the heat release then helps to sustain the decomposition. The purpose here is not to consider the broad field of catalysis, but simply point out where ignition is important with regard to exothermic decomposition both with regard to sustained decomposition and consideration of safety in handling of such chemicals. Wolfe [22] found that copper, chromium, manganese,

Ignition

407

nickel, and iron enhanced hydrazine decomposition and that cadmium, zinc, magnesium, and aluminum did not. If one examines the electronic structure of these metals, one will note that non-catalysts have either no d sub-shells or complete d sub-shell orbitals [21]. In hydrazine decomposition ignition on the catalytic surfaces mentioned, one step seems to form bonds between the N atoms in hydrazine and incomplete d-orbitals of the metal. This bonding initiates dissociation of the hydrazine and subsequent decomposition and heat release. It is generally believed then that with metals the electronic configuration, in particular the d-band is an index of catalytic activity [21]. In this theory it is believed that in the absorption of the gas on the metal surface, electrons are donated by the gas to the d-band of the metal, thus filling the fractional deficiencies or holes in the d-band. Obviously, noble metal surfaces are particularly best for catalytic initiation or ignition, as they do not have the surface oxide layer formation discussed in the previous sections. In catalytic combustion or any exothermic decomposition, thermal aspects can dominate the continued reacting process particularly since the catalytic surface will also rise in temperature.

PROBLEMS (Those with an asterisk require a numerical solution) 1. The reported decomposition of ammonium nitrate indicates that the reaction is unimolecular and that the rate constant has an A factor of 1013.8 and an activation energy of 170 kJ/mol. Using this information, determine the critical storage radius at 160°C. Report the calculation so that a plot of rcrit versus T0 can be obtained. Take a temperature range from 80°C to 320°C. 2. Concisely explain the difference between chain and thermal explosions. 3. Are liquid droplet ignition times appreciably affected by droplet size? Explain. 4. The critical size for the storage of a given solid propellant at a temperature of 325 K is 4.0 m. If the activation energy of the solid reacting mixture were 185 kJ/mol, what critical size would hold for 340 K. 5. A fuel-oxidizer mixture at a given temperature To  550K ignites. If the overall activation energy of the reaction is 240 kJ/mol, what is the true ignition temperature Ti? How much faster is the reaction at Ti compared to that at To? What can you say about the difference between Ti and To for a very large activation energy process? 6. Verify the result in Table 1 for the condition where the shock and compression ratio is 5. Consider the gas to be air in which the ratio of specific heats is 1.385. 7.* Calculate the ignition delays of a dilute H2/O2 mixture in Ar as a function of initial mixture temperature at a pressure of 5 atm for a constant pressure adiabatic system. The initial mixture consists of H2 with a mole fraction of 0.01 and O2 with a mole fraction of 0.005. The balance of the mixture is argon. Make a plot of temperature versus time for a mixture initially at 1100 K and

408

Combustion

determine a criteria for determining ignition delays. Also plot the species moles fractions of H2, O2, H2O, OH, H, O, HO2, and H2O2 as a function of time for this condition. Discuss the behavior of the species profiles. Plot the sensitivities of the temperature profile to variations in the A factors of each of the reactions and identify the two most important reactions during the induction period and reaction. Determine the ignition delays at temperatures of 850, 900, 950, 1000, 1050, 1100, 1200, 1400, and 1600 K and make a plot of ignition delay versus inverse temperature. Describe the trend observed in the ignition delay profile. How is it related to the H2/O2 explosion limits? What is the overall activation energy?

REFERENCES 1. Mullins, H. P., “Spontaneous Ignition of Liquid Fuels.” Chap. 11, Agardograph No. 4. Butterworth, London, 1955. 2. Zabetakis, M. G., Bull. US Bur. Mines No. 627 (1965). 3. Semenov, N. N., “Chemical Kinetics and Chain Reactions.” Oxford University Press, London, 1935. 4. Hinshelwood, C. N., “The Kinetics of Chemical Change.” Oxford University Press, London, 1940. 5. Zeldovich, Y. B., Barenblatt, G. I., Librovich, V. B., and Makhviladze, G. M., “The Mathematical Theory of Combustion and Explosions.” Chap. 1. Consultants Bureau, New York, 1985. 6. Frank-Kamenetskii, D., “Diffusion and Heat Exchange in Chemical Kinetics.” Princeton University Press, Princeton, NJ, 1955. 7. Hainer, R., Proc. Combust. Inst. 5, 224 (1955). 7a. Kipping, F. S., and Kipping, F. B., “Organic Chemistry,” Chap. XV. Crowell Co., New York, 1941. 8. Mullins, B. P., Fuel 32, 343 (1953). 9. Mullins, B. P., Fuel 32, 363 (1953). 10. Todes, O. M., Acta Physicochem. URRS 5, 785 (1936). 11. Jost, W., “Explosions and Combustion Processes in Gases.” Chap. 1. McGraw-Hill, New York, 1946. 12. Belles, F. E., and Swett, C. C., Natl. Advis. Comm. Aeronaut. Rep. No. 1300 (1989). 13. Zeldovich, Y B., Zh. Eksp. Teor. Fiz., 11, 159 (1941) see Shchelinkov, Y S., “The Physics of the Combustion of Gases,” Chap. 5. Edited Transl. FTD-HT-23496-68, Transl. Revision, For. Technol. Div., Wright-Patterson AFB, Ohio, 1969. 14. Blanc, M. V., Guest, P. G., von Elbe, G., and Lewis, B., Proc. Combust. Inst. 3, 363 (1949). 15. Calcote, H. F., Gregory, C. A. Jr., Barnett, C. M., and Gilmer, R. B., Ind. Eng. Chem. 44, 2656 (1952). 16. Kanury, A. M., “Introduction to Combustion Phenomena.” Chap. 4. Gordon & Breach, New York, 1975. 17. Hertzberg, M., Cashdollar, K. L., Contic, R. S., and Welsch, L. M., Bur. Mines Rep. (1984). 18. Dryer, F. L., and Glassman, I., Prog. Astronaut. Aeronaut. 26, 255 (1978). 19. “The Random House Dictionary of the English Language”, Random House, New York, 1967. 20. Babrauskas, V., “Ignition Handbook.” Fire Science Publishers, Issaquah, WA, 2003. 21. Eberstein, I., and Glassman, I., “Prog. in Astronaut Rocketry,” Vol. 2. 1960, Academic Press, New York, p.351. 22. W. Wolfe, Engineering Report, 126, Olin Mathieson, 1953. 23. Glassman, I., Papas, P., and Brezinsky, K., Combust. Sci. Technol. 83, 161 (1992).

Chapter 8

Environmental Combustion Considerations A. INTRODUCTION In the mid 1940s, symptoms now attributable to photochemical air pollution were first encountered in the Los Angeles area. Several researchers recognized that the conditions there were producing a new kind of smog caused by the action of sunlight on the oxides of nitrogen and subsequent reactions with hydrocarbons. This smog was different from the “pea-soup” conditions prevailing in London in the early twentieth century and the polluted-air disaster that struck Donora, Pennsylvania, in the 1930s. It was also different from the conditions revealed by the opening of Eastern Europe in the last part of the 20th century. In Los Angeles, the primary atmosphere source of nitrogen oxides, CO, and hydrocarbons was readily shown to be the result of automobile exhausts. The burgeoning population and industrial growth in US urban and exurban areas were responsible for the problem of smog, which led to controls not only on automobiles, but also on other mobile and stationary sources. Atmospheric pollution has become a worldwide concern. With the prospect of supersonic transports flying in the stratosphere came initial questions as to how the water vapor ejected by the power plants of these planes would affect the stratosphere. This concern led to the consideration of the effects of injecting large amounts of any species on the ozone balance in the atmosphere. It then became evident that the major species that would affect the ozone balance were the oxides of nitrogen. The principal nitrogen oxides found to be present in the atmosphere are nitric oxide (NO) and nitrogen dioxide (NO2)—the combination of which is referred to as NOx—and nitrous oxide (N2O). As Bowman [1] has reported, the global emissions of NOx and N2O into the atmosphere have been increasing steadily since the middle of the nineteenth century. And, although important natural sources of the oxides of nitrogen exist, a significant amount of this increase is attributed to human activities, particularly those involving combustion of fossil and biomass fuels. For details as to the sources of combustion-generated nitrogen oxide emissions, one should refer to Bowman’s review [1]. Improvement of the atmosphere continues to be of great concern. The continual search for fossil fuel resources can lead to the exploitation of coal, shale, 409

410

Combustion

and secondary and tertiary oil recovery schemes. For instance, the industrialization of China, with its substantial resource of sulfur coals, requires consideration of the effect of sulfur oxide emissions. Indeed, the sulfur problem may be the key in the more rapid development of coal usage worldwide. Furthermore, the fraction of aromatic compounds in liquid fuels derived from such natural sources or synthetically developed is found to be large, so that, in general, such fuels have serious sooting characteristics. This chapter seeks not only to provide better understanding of the oxidation processes of nitrogen and sulfur and the processes leading to particulate (soot) formation, but also to consider appropriate combustion chemistry techniques for regulating the emissions related to these compounds. The combustion— or, more precisely, the oxidation—of CO and aromatic compounds has been discussed in earlier chapters. This information and that to be developed will be used to examine the emission of other combustion-generated compounds thought to have detrimental effects on the environment and on human health. How emissions affect the atmosphere is treated first.

B. THE NATURE OF PHOTOCHEMICAL SMOG Photochemical air pollution consists of a complex mixture of gaseous pollutants and aerosols, some of which are photochemically produced. Among the gaseous compounds are the oxidizing species ozone, nitrogen dioxide, and peroxyacyl nitrate: O R O3

NO2

Ozone

Nitrogen dioxide

C OONO2

Peroxyacyl nitrate

The member of this series most commonly found in the atmosphere is peroxyacetyl nitrate (PAN) O H3C

C OONO2

Peroxyacetyl nitrate (PAN)

The three compounds O3, NO2, and PAN are often grouped together and called photochemical oxidant. Photochemical smog comprises mixtures of particulate matter and noxious gases, similar to those that occurred in the typical London-type “peasoup” smog. The London smog was a mixture of particulates and oxides of sulfur, chiefly sulfur dioxide. But the overall system in the London smog was

411

Environmental Combustion Considerations

chemically reducing in nature. This difference in redox chemistry between photochemical oxidant and SOx-particulate smog is quite important in several respects. Note in particular the problem of quantitatively detecting oxidant in the presence of sulfur dioxide. Being a reducing agent SOx tends to reduce the oxidizing effects of ozone and thus produces low quantities of the oxidant. In dealing with the heterogeneous gas–liquid–solid mixture characterized as photochemical smog, it is important to realize from a chemical, as well as a biological, point of view that synergistic effects may occur.

1. Primary and Secondary Pollutants Primary pollutants are those emitted directly to the atmosphere while secondary pollutants are those formed by chemical or photochemical reactions of primary pollutants after they have been admitted to the atmosphere and exposed to sunlight. Unburned hydrocarbons, NO, particulates, and the oxides of sulfur are examples of primary pollutants. The particulates may be lead oxide from the oxidation of tetraethyllead in automobiles, fly ash, and various types of carbon formation. Peroxyacyl nitrate and ozone are examples of secondary pollutants. Some pollutants fall in both categories. Nitrogen dioxide, which is emitted directly from auto exhaust, is also formed in the atmosphere photochemically from NO. Aldehydes, which are released in auto exhausts, are also formed in the photochemical oxidation of hydrocarbons. Carbon monoxide, which arises primarily from autos and stationary sources, is likewise a product of atmospheric hydrocarbon oxidation.

2. The Effect of NOX It has been well established that if a laboratory chamber containing NO, a trace of NO2, and air is irradiated with ultraviolet light, the following reactions occur:



NO2  hv (3000 A  λ  4200 A) → NO  O (3 P)

(8.1)

O  O 2  M → O3  M

(8.2)

O3  NO → O2  NO2

(8.3)

The net effect of irradiation on this inorganic system is to establish the dynamic equilibrium hv NO2  O2 ←⎯⎯ → NO  O3

(8.4)

412

Combustion

However, if a hydrocarbon, particularly an olefin or an alkylated benzene, is added to the chamber, the equilibrium represented by reaction (8.4) is unbalanced and the following events take place: 1. 2. 3. 4.

The hydrocarbons are oxidized and disappear. Reaction products such as aldehydes, nitrates, PAN, etc., are formed. NO is converted to NO2. When all the NO is consumed, O3 begins to appear. On the other hand, PAN and other aldehydes are formed from the beginning.

Basic rate information permits one to examine these phenomena in detail. Leighton [2], in his excellent book Photochemistry of Air Pollution, gives numerous tables of rates and products of photochemical nitrogen oxide– hydrocarbon reactions in air; this early work is followed here to give fundamental insight into the photochemical smog problem. The data in these tables show low rates of photochemical consumption of the saturated hydrocarbons, as compared to the unsaturates, and the absence of aldehydes in the products of the saturated hydrocarbon reactions. These data conform to the relatively low rate of reaction of the saturated hydrocarbons with oxygen atoms and their inertness with respect to ozone. Among the major products in the olefin reactions are aldehydes and ketones. Such results correspond to the splitting of the double bond and the addition of an oxygen atom to one end of the olefin. Irradiation of mixtures of an olefin with nitric oxide and nitrogen dioxide in air shows that the nitrogen dioxide rises in concentration before it is eventually consumed by reaction. Since it is the photodissociation of the nitrogen dioxide that initiates the reaction, it would appear that a negative quantum yield results. More likely, the nitrogen dioxide is being formed by secondary reactions more rapidly than it is being photodissociated. The important point is that this negative quantum yield is realized only when an olefin (hydrocarbon) is present. Thus, adding the overall step O ⎫⎪  olefin → products O3 ⎬⎪⎪⎭

(8.5)

to reactions (8.1)–(8.3) would not be an adequate representation of the atmospheric photochemical reactions. However, if one assumes that O3 attains a steady-state concentration in the atmosphere, then one can perform a steadystate analysis (see Chapter 2, Section B) with respect to O3. Furthermore, if one assumes that O3 is largely destroyed by reaction (8.3), one obtains a very useful approximate relationship: (O3 )  ( j1 /k3 ) (NO2 )/ (NO) where j is the rate constant for the photochemical reaction. Thus, the O3 steady-state concentration in a polluted atmosphere is seen to increase with

Environmental Combustion Considerations

413

decreasing concentration of nitric oxide and vice versa. The ratio of j1/k3 approximately equals 1.2 ppm for the Los Angeles noonday condition [2]. Reactions such as O  NO2 → NO  O2 O  NO2  M → NO3  M NO3  NO → 2 NO2 O  NO  M → NO2  M 2 NO  O2 → 2 NO2 NO3  NO2 → N 2 O5 N 2 O5 → NO3  NO2 do not play a part. They are generally too slow to be important. Furthermore, it has been noted that when the rate of the oxygen atom– olefin reaction and the rate of the ozone–olefin reaction are totaled, they do not give the complete hydrocarbon consumption. This anomaly is also an indication of an additional process. An induction period with respect to olefin consumption is also observed in the photochemical laboratory experiments, thus indicating the buildup of an intermediate. When illumination is terminated in these experiments, the excess rate over the total of the O and O3 reactions disappears. These and other results suggest that the intermediate formed is photolyzed and contributes to the concentration of the major species of concern. Possible intermediates that fulfill the requirements of the laboratory experiments are alkyl and acyl nitrites and pernitrites. The second photolysis effect eliminates the possibility that aldehydes serve as the intermediate. Various mechanisms have been proposed to explain the aforementioned laboratory results. The following low-temperature (atmospheric) sequence based on isobutene as the initial fuel was first proposed by Leighton [2] and appears to account for most of what has been observed: O  C4 H8 → CH3  C3 H 5 O

(8.6)

CH3  O2 → CH3 OO

(8.7)

CH3 OO  O2 → CH3 O  O3

(8.8)

O3  NO → NO2  O2

(8.9)

CH3 O  NO → CH3 ONO

(8.10)

CH3 ONO  hv → CH3 O*  NO

(8.11)

CH3 O*  O2 → H 2 CO  HOO

(8.12)

414

Combustion

HOO  C4 H8 → H 2 CO  (CH3 )2 CO  H

(8.13)

M  H  O2 → HOO  M

(8.14)

HOO  NO → OH  NO2

(8.15)

OH  C4 H8 → (CH3 )2 CO  CH3

(8.16)

CH 3  O2 → CH 3 OO (as above)

(8.17)

2HOO → H 2 O2  O2

(8.18)

2OH → H 2  O2

(8.19)

HOO  H 2 → H 2 O  OH

(8.20)

HOO  H 2 → H 2 O2  H

(8.21)

There are two chain-propagating sequences [reactions (8.13) and (8.14) and reactions (8.15)–(8.17)] and one chain-breaking sequence [reactions (8.18) and (8.19)]. The intermediate is the nitrite as shown in reaction (8.10). Reaction (8.11) is the required additional photochemical step. For every NO2 used to create the O atom of reaction (8.6), one is formed by reaction (8.9). However, reactions (8.10), (8.11), and (8.15) reveal that for every two NO molecules consumed, one NO and one NO2 form—hence the negative quantum yield of NO2. With other olefins, other appropriate reactions may be substituted. Ethylene would give O  C2 H 4 → CH3  HCO

(8.22)

HOO  C2 H 4 → 2H 2 CO  H

(8.23)

OH  C2 H 4 → H 2 CO  CH3

(8.24)

OH  C3 H6 → CH3 CHO  CH3

(8.25)

Propylene would add

Thus, PAN would form from hv CH3 CHO  O2 ⎯ ⎯⎯ → CH3 CO  HOO

(8.26)

CH3 CO  O2 → CH3 (CO)OO

(8.27)

CH3 (CO)OO  NO2 → CH3 (CO)OONO2

(8.28)

and an acid could form from the overall reaction CH3 (CO)OO  2CH3 CHO → CH3 (CO)OH  2CH3 CO  OH (8.29)

Environmental Combustion Considerations

415

Since pollutant concentrations are generally in the parts-per-million range, it is not difficult to postulate many types of reactions and possible products.

3. The Effect of SOX Historically, the sulfur oxides have long been known to have a deleterious effect on the atmosphere, and sulfuric acid mist and other sulfate particulate matter are well established as important sources of atmospheric contamination. However, the atmospheric chemistry is probably not as well understood as the gas-phase photoxidation reactions of the nitrogen oxides–hydrocarbon system. The pollutants form originally from the SO2 emitted to the air. Just as mobile and stationary combustion sources emit some small quantities of NO2 as well as NO, so do they emit some small quantities of SO3 when they burn sulfurcontaining fuels. Leighton [2] also discusses the oxidation of SO2 in polluted atmospheres and an excellent review by Bulfalini [3] has appeared. This section draws heavily from these sources. The chemical problem here involves the photochemical and catalytic oxidation of SO2 and its mixtures with the hydrocarbons and NO; however the primary concern is the photochemical reactions, both gas-phase and aerosol-forming. The photodissociation of SO2 into SO and O atoms is markedly different from the photodissociation of NO2. The bond to be broken in the sulfur compound requires about 560 kJ/mol. Thus, wavelengths greater than 2180 Å do not have sufficient energy to initiate dissociation. This fact is significant in that only solar radiation greater than 2900 Å reaches the lower atmosphere. If a photochemical effect is to occur in the SO2–O2 atmospheric system, it must be that the radiation electronically excites the SO2 molecule but does not dissociate it. There are two absorption bands of SO2 within the range 3000–4000 Å. The first is a weak absorption band and corresponds to the transition to the first excited state (a triplet). This band originates at 3880 Å and has a maximum around 3840 Å. The second is a strong absorption band and corresponds to the excitation to the second excited state (a triplet). This band originates at 3376 Å and has a maximum around 2940 Å. Blacet [4], who carried out experiments in high O2 concentrations, reported that ozone and SO3 appear to be the only products of the photochemically induced reaction. The following essential steps were postulated: SO2  hv → SO*2

(8.30)

SO*2  O2 → SO 4

(8.31)

SO 4  O2 → SO3  O3

(8.32)

The radiation used was at 3130 Å, and it would appear that the excited SO*2 in reaction (8.30) is a singlet. The precise roles of the excited singlet and

416

Combustion

triplet states in the photochemistry of SO2 are still unclear [3]. Nevertheless, this point need not be one of great concern since it is possible to write the reaction sequence

1 SO* 2

SO2  hv → 1 SO*2

(8.33)

 SO2 → 3 SO*2  SO2

(8.34)

Thus, reaction (8.30) could specify either an excited singlet or triplet SO*2 . The excited state may, of course, degrade by internal transfer to a vibrationally excited ground state that is later deactivated by collision, or it may be degraded directly by collisions. Fluorescence of SO2 has not been observed above 2100 Å. The collisional deactivation steps known to exist in laboratory experiments are not listed here in order to minimize the writing of reaction steps. Since they involve one species in large concentrations, reactions (8.30)–(8.32) are the primary ones for the photochemical oxidation of SO2 to SO3. A secondary reaction route to SO3 could be SO 4  SO2 → 2SO3

(8.35)

In the presence of water a sulfuric acid mist forms according to H 2 O  SO3 → H 2 SO 4

(8.36)

The SO4 molecule formed by reaction (8.31) would probably have a peroxy structure; and if SO*2 were a triplet, it might be a biradical. There is conflicting evidence with respect to the results of the photolysis of mixtures of SO2, NOx, and O2. However, many believe that the following should be considered with the NOx photolysis reactions: SO2  NO → SO  NO2

(8.37)

SO2  NO2 → SO3  NO

(8.38)

SO2  O  M → SO3  M

(8.39)

SO2  O3 → SO3  O2

(8.40)

SO3  O → SO2  O2

(8.41)

SO 4  NO → SO3  NO2

(8.42)

SO 4  NO2 → SO3  NO3

(8.43)

SO 4  O → SO3  O2

(8.44)

Environmental Combustion Considerations

417

SO  O  M → SO2  M

(8.45)

SO  O3 → SO2  O2

(8.46)

SO  NO2 → SO2  NO

(8.47)

The important reducing effect of the SO2 with respect to different polluted atmospheres mentioned in the introduction of this section becomes evident from these reactions. Some work [5] has been performed on the photochemical reaction between sulfur dioxide and hydrocarbons, both paraffins and olefins. In all cases, mists were found, and these mists settled out in the reaction vessels as oils with the characteristics of sulfuric acids. Because of the small amounts of materials formed, great problems arise in elucidating particular steps. When NOx and O2 are added to this system, the situation is most complex. Bulfalini [3] sums up the status in this way: “The aerosol formed from mixtures of the lower hydrocarbons with NOx and SO2 is predominantly sulfuric acid, whereas the higher olefin hydrocarbons appear to produce carbonaceous aerosols also, possibly organic acids, sulfonic or sulfuric acids, nitrate-esters, etc.”

C. FORMATION AND REDUCTION OF NITROGEN OXIDES The previous sections help establish the great importance of the nitrogen oxides in the photochemical smog reaction cycles described. Strong evidence indicated that the major culprit in NOx production was the automobile. But, as automobile emissions standards were enforced, attention was directed to power generation plants that use fossil fuels. Given these concerns and those associated with supersonic flight in the stratosphere, great interest remains in predicting—and reducing—nitrogen oxide emissions; this interest has led to the formulation of various mechanisms and analytical models to predict specifically the formation and reduction of nitrogen oxides in combustion systems. This section offers some insight into these mechanisms and models, drawing heavily from the reviews by Bowman [1] and Miller and Bowman [6]. When discussing nitrogen oxide formation from nitrogen in atmospheric air, one refers specifically to the NO formed in combustion systems in which the original fuel contains no nitrogen atoms chemically bonded to other chemical elements such as carbon or hydrogen. Since this NO from atmospheric air forms most extensively at high temperatures, it is generally referred to as thermal NO. One early controversy with regard to NOx chemistry revolved around what was termed “prompt” NO. Prompt NO was postulated to form in the flame zone by mechanisms other than those thought to hold exclusively for NO formation from atmospheric nitrogen in the high-temperature zone of the flame or post-flame zone. Although the amount of prompt NO formed is quite

418

Combustion

small under most practical conditions, the fundamental studies into this problem have helped clarify much about NOx formation and reduction both from atmospheric and fuel-bound nitrogen. The debate focused on the question of whether prompt NO formation resulted from reaction of hydrocarbon radicals and nitrogen in the flame or from nitrogen reactions with large quantities of O atoms generated early in the flame. Furthermore, it was suggested that superequilibrium concentrations of O atoms could, under certain conditions of pressure and stoichiometry, lead to the formation of nitrous oxide, N2O, a subsequent source of NO. These questions are fully addressed later in this section. The term “prompt” NO derives from the fact that the nitrogen in air can form small quantities of CN compounds in the flame zone. In contrast, thermal NO forms in the high-temperature post-flame zone. These CN compounds subsequently react to form NO. The stable compound HCN has been found in the flame zone and is a product in very fuel-rich flames. Chemical models of hydrocarbon reaction processes reveal that, early in the reaction, O atom concentrations can reach superequilibrium proportions; and, indeed, if temperatures are high enough, these high concentrations could lead to early formation of NO by the same mechanisms that describe thermal NO formation. NOx formation from fuel-bound nitrogen is meant to specify, as mentioned, the nitrogen oxides formed from fuel compounds that are chemically bonded to other elements. Fuel-bound nitrogen compounds are ammonia, pyridine, and many other amines. The amines can be designated as RNH2, where R is an organic radical or H atom. The NO formed from HCN and the fuel fragments from the nitrogen compounds are sometimes referred to as chemical NO in terminology analogous to that of thermal NO. Although most early analytical and experimental studies focused on NO formation, more information now exists on NO2 and the conditions under which it is likely to form in combustion systems. Some measurements in practical combustion systems have shown large amounts of NO2, which would be expected under the operating conditions. Controversy has surrounded the question of the extent of NO2 formation in that the NO2 measured in some experiments may actually have formed in the probes used to capture the gas sample. Indeed, some recent high-pressure experiments have revealed the presence of N2O.

1. The Structure of the Nitrogen Oxides Many investigators have attempted to investigate analytically the formation of NO in fuel–air combustion systems. Given of the availability of an enormous amount of computer capacity, they have written all the reactions of the nitrogen oxides they thought possible. Unfortunately, some of these investigators have ignored the fact that some of the reactions could have been eliminated because of steric considerations, as discussed with respect to sulfur oxidation. Since the structure of the various nitrogen oxides can be important, their formulas and structures are given in Table 8.1.

419

Environmental Combustion Considerations

TABLE 8.1 Structure of Gaseous Nitrogen Compounds Nitrogen N2

N

Nitrous oxide N2O

N

Nitric oxide NO

N

N 

O

N



N



N

O

O

Nitrogen dioxide NO2

O 140°

N O

 O

Nitrate ion NO3 

N

Nitrogen tetroxide N2O4

O  O O

O N



N



O Nitrogen pentoxide

O N

or

O  N

N

O

O

O O

O O

O

N

O

O  N

O O

N

O O

2. The Effect of Flame Structure As the important effect of temperature on NO formation is discussed in the following sections, it is useful to remember that flame structure can play a most significant role in determining the overall NOx emitted. For premixed systems like those obtained on Bunsen and flat flame burners and almost obtained in carbureted spark-ignition engines, the temperature, and hence the mixture ratio, is the prime parameter in determining the quantities of NOx formed. Ideally, as in equilibrium systems, the NO formation should peak at the stoichiometric value and decline on both the fuel-rich and fuel-lean sides, just as the temperature does. Actually, because of kinetic (nonequilibrium) effects, the peak is found somewhat on the lean (oxygen-rich) side of stoichiometric.

420

Combustion

However, in fuel-injection systems where the fuel is injected into a chamber containing air or an air stream, the fuel droplets or fuel jets burn as diffusion flames, even though the overall mixture ratio may be lean and the final temperature could correspond to this overall mixture ratio. The temperature of these diffusion flames is at the stoichiometric value during part of the burning time, even though the excess species will eventually dilute the products of the flame to reach the true equilibrium final temperature. Thus, in diffusion flames, more NOx forms than would be expected from a calculation of an equilibrium temperature based on the overall mixture ratio. The reduction reactions of NO are so slow that in most practical systems the amount of NO formed in diffusion flames is unaffected by the subsequent drop in temperature caused by dilution of the excess species.

3. Reaction Mechanisms of Oxides of Nitrogen Nitric oxide is the primary nitrogen oxide emitted from most combustion sources. The role of nitrogen dioxide in photochemical smog has already been discussed. Stringent emission regulations have made it necessary to examine all possible sources of NO. The presence of N2O under certain circumstances could, as mentioned, lead to the formation of NO. In the following subsections the reaction mechanisms of the three nitrogen oxides of concern are examined.

a. Nitric Oxide Reaction Mechanisms There are three major sources of the NO formed in combustion: (1) oxidation of atmospheric (molecular) nitrogen via the thermal NO mechanisms; (2) prompt NO mechanisms; and (3) oxidation of nitrogen-containing organic compounds in fossil fuels via the fuel-bound NO mechanisms [1]. The extent to which each contributes is an important consideration. Thermal NO mechanisms: For premixed combustion systems a conservative estimate of the thermal contribution to NO formation can be made by consideration of the equilibrium system given by reaction (8.48): N 2  O2  2 NO

(8.48)

As is undoubtedly apparent, the kinetic route of NO formation is not the attack of an oxygen molecule on a nitrogen molecule. Mechanistically, as described in Chapter 3, oxygen atoms form from the H2 ¶O2 radical pool, or possibly from the dissociation of O2, and these oxygen atoms attack nitrogen molecules to start the simple chain shown by reactions (8.49) and (8.50): O  N 2  NO  N

kf  2  1014 exp(315 /RT )

(8.49)

N  2  NO  O

kf  6.4  109 exp (26 /RT )

(8.50)

421

Environmental Combustion Considerations

where the activation energies are in kJ/mol. Since this chain was first postulated by Zeldovich [7], the thermal mechanism is often referred to as the Zeldovich mechanism. Common practice now is to include the step N  OH  NO  H

kf  3.8  1013

(8.51)

in the thermal mechanism, even though the reacting species are both radicals and therefore the concentration terms in the rate expression for this step would be very small. The combination of reactions (8.49)–(8.51) is frequently referred to as the extended Zeldovich mechanism. If one invokes the steady-state approximation described in Chapter 2 for the N atom concentration and makes the partial equilibrium assumption also described in Chapter 2 for the reaction system H  O2  OH  O one obtains for the rate of formation of NO [8] ⎪⎧ 1  [(NO)2 /K (O2 )(N 2 )] ⎪⎫⎪ d (NO)  2 k49f (O)(N 2 ) ⎪⎨ ⎬ ⎪⎪ 1  [k49 b (NO)/k50 f (O2 )] ⎪⎪ dt ⎩ ⎭ 2 K   K 49 /K 50  K c,f,NO

(8.52)

where K is the concentration equilibrium constant for the specified reaction system and K the square of the equilibrium constant of formation of NO. In order to calculate the thermal NO formation rate from the preceding expression, it is necessary to know the concentrations of O2, N2, O, and OH. But the characteristic time for the forward reaction (8.49) always exceeds the characteristic times for the reaction systems that make up the processes in fuel–oxidizer flame systems; thus, it would appear possible to decouple the thermal NO process from the flame process. Using such an assumption, the NO formation can be calculated from Eq. (8.52) using local equilibrium values of temperature and concentrations of O2, N2, O, and OH. From examination of Eq. (8.52), one sees that the maximum NO formation rate is given by d (NO) / dt  2 k49f (O)(N 2 )

(8.53)

which corresponds to the condition that (NO)

(NO)eq. Due to the assumed equilibrium condition, the concentration of O atoms can be related to the concentration of O2 molecules via 1 2

O2  O

K c,f,O,Teq  (O)eq / (O2 )1eq/2

422

Combustion

and Eq. (8.53) becomes d (NO)/dt  2 k49f K c,f,O,Teq (O2 )1eq/2 (N 2 )eq

(8.54)

6

1.0

5

0.8

4

0.6

3

0.4

2

0.2

1 1600

1800

2000

2200

2400

[NO]/[NO]eq.

log {d [NO]/dtmax/d[NO]/dtZeldovich}

The strong dependence of thermal NO formation on the combustion temperature and the lesser dependence on the oxygen concentration are evident from Eq. (8.54). Thus, considering the large activation energy of reaction (8.49), the best practical means of controlling NO is to reduce the combustion gas temperature and, to a lesser extent, the oxygen concentration. For a condition of constant temperature and varying pressure Eq. (8.53) suggests that the O atom concentration will decrease as the pressure is raised according to Le Chatelier’s principle and the maximum rate will decrease. Indeed, this trend is found in fluidized bed reactors. In order to determine the errors that may be introduced by the Zeldovich model, Miller and Bowman [6] calculated the maximum (initial) NO formation rates from the model and compared them with the maximum NO formation rates calculated from a detailed kinetics model for a fuel-rich (φ  1.37) methane–air system. To allow independent variation of temperature, an isothermal system was assumed and the type of prompt NO reactions to be discussed next were omitted. Thus, the observed differences in NO formation rates are due entirely to the nonequilibrium radical concentrations that exist during the combustion process. Their results are shown in Fig. 8.1, which indicates

0 2600

Temperature (K)

FIGURE 8.1 The effect of superequilibrium radical concentrations on NO formation rates in the isothermal reaction of 13% methane in air (φ  1.37). The upper curve is the ratio of the maximum NO formation rate calculated using the detailed reaction mechanism of Ref. [6] to the initial NO formation rate calculated using the Zeldovich model. The lower curve is the ratio of the NO concentration at the time of the maximum NO formation rate calculated using the detailed reaction mechanism to the equilibrium NO concentration (from Miller and Bowman [6]).

Environmental Combustion Considerations

423

a noticeable acceleration of the maximum NO formation rate above that calculated using the Zeldovich model during the initial stages of the reaction due to nonequilibrium effects, with the departures from the Zeldovich model results decreasing with increasing temperature. As the lower curve in Fig. 8.1 indicates, while nonequilibrium effects are evident over a wide range of temperature, the accelerated rates are sufficiently low that very little NO is formed by the accelerated nonequilibrium component. Examining the lower curve, as discussed in Chapter 1, one sees that most hydrocarbon–air combustion systems operate in the range of 2100–2600 K. Prompt NO mechanisms: In dealing with the presentation of prompt NO mechanisms, much can be learned by considering the historical development of the concept of prompt NO. With the development of the Zeldovich mechanism, many investigators followed the concept that in premixed flame systems, NO would form only in the post-flame or burned gas zone. Thus, it was thought possible to experimentally determine thermal NO formation rates and, from these rates, to find the rate constant of Eq. (8.49) by measurement of the NO concentration profiles in the post-flame zone. Such measurements can be performed readily on flat flame burners. Of course, in order to make these determinations, it is necessary to know the O atom concentrations. Since hydrocarbon–air flames were always considered, the nitrogen concentration was always in large excess. As discussed in the preceding subsection, the O atom concentration was taken as the equilibrium concentration at the flame temperature and all other reactions were assumed very fast compared to the Zeldovich mechanism. These experimental measurements on flat flame burners revealed that when the NO concentration profiles are extrapolated to the flame-front position, the NO concentration goes not to zero, but to some finite value. Such results were most frequently observed with fuel-rich flames. Fenimore [9] argued that reactions other than the Zeldovich mechanism were playing a role in the flame and that some NO was being formed in the flame region. He called this NO, “prompt” NO. He noted that prompt NO was not found in nonhydrocarbon CO–air and H2–air flames, which were analyzed experimentally in the same manner as the hydrocarbon flames. The reaction scheme he suggested to explain the NO found in the flame zone involved a hydrocarbon species and atmospheric nitrogen. The nitrogen compound was formed via the following mechanism: CH  N 2  HCN  N

(8.55)

C2  N 2  2CN

(8.56)

The N atoms could form NO, in part at least, by reactions (8.50) and (8.51), and the CN could yield NO by oxygen or oxygen atom attack. It is well known that CH exists in flames and indeed, as stated in Chapter 4, is the molecule that gives the deep violet color to a Bunsen flame.

424

Combustion

100

N2

Mole fraction, X

101

CO2 CO H2 & H2O [NO] eq. OH H

102 103

O2 O

104 105

NO

N 106 7 6 5 4 3 10 10 10 10 10 102 Time (s) FIGURE 8.2 Concentration–time profiles in the kinetic calculation of the methane–air reaction at an inlet temperature of 1000 K. P2  10 atm, φ  1.0, and Teq  2477 K (from Martenay [11]).

In order to verify whether reactions other than the Zeldovich mechanism were effective in NO formation, various investigators undertook the study of NO formation kinetics by use of shock tubes. The primary work in this area was that of Bowman and Seery [10] who studied the CH4¶O2¶N2 system. Complex kinetic calculations of the CH4¶O2¶N2 reacting system based on early kinetic rate data at a fixed high temperature and pressure similar to those obtained in a shock tube [11] for T  2477 K and P  10 atm are shown in Fig. 8.2. Even though more recent kinetic rate data would modify the product– time distribution somewhat, it is the general trends of the product distribution which are important and they are relatively unaffected by some changes in rates. These results are worth considering in their own right, for they show explicitly much that has been implied. Examination of Fig. 8.2 shows that at about 5  105 s, all the energy-release reactions will have equilibrated before any significant amounts of NO have formed; and, indeed, even at 102 s the NO has not reached its equilibrium concentration for T  2477 K. These results show that for such homogeneous or near-homogeneous reacting systems, it would be possible to quench the NO reactions, obtain the chemical heat release, and prevent NO formation. This procedure has been put in practice in certain combustion schemes. Equally important is the fact that Fig. 8.2 reveals large overshoots within the reaction zone. If these occur within the reaction zone, the O atom concentration could be orders of magnitude greater than its equilibrium value, in which case this condition could lead to the prompt NO found in flames. The mechanism analyzed to obtain the results depicted in Fig. 8.2 was essentially that given in Chapter 3 Section G2 with the Zeldovich reactions. Thus it was thought possible that the Zeldovich mechanism could account for the prompt NO.

425

Environmental Combustion Considerations

NO conc (108 mol/cm3)

1.6 equil. φ  0.5

1.2

φ  0.5 lean equil. φ  1.2

0.8 φ  1.2 rich

0.4

0

0

0.2

0.4

0.6

0.8

1.0

Time (ms) FIGURE 8.3 Comparison of measured and calculated NO concentration profiles for a CH4¶O2¶N2 mixture behind reflected shocks. Initial post-shock conditions: T  2960 K, P  3.2 atm (from Bowman [12]).

The early experiments of Bowman and Seery appeared to confirm this conclusion. Some of their results are shown in Fig. 8.3. In this figure the experimental points compared very well with the analytical calculations based on the Zeldovich mechanisms alone. The same computational program as that of Martenay [11] was used. Figure 8.3 also depicts another result frequently observed: fuel-rich systems approach NO equilibrium much faster than do fuel-lean systems [12]. Although Bowman and Seery’s results would, at first, seem to refute the suggestion by Fenimore that prompt NO forms by reactions other than the Zeldovich mechanism, one must remember that flames and shock tubeinitiated reacting systems are distinctively different processes. In a flame there is a temperature profile that begins at the ambient temperature and proceeds to the flame temperature. Thus, although flame temperatures may be simulated in shock tubes, the reactions in flames are initiated at much lower temperatures than those in shock tubes. As stressed many times before, the temperature history frequently determines the kinetic route and the products. Therefore shock tube results do not prove that the Zeldovich mechanism alone determines prompt NO formation. The prompt NO could arise from other reactions in flames, as suggested by Fenimore. Bachmeier et al. [13] appear to confirm Fenimore’s initial postulates and to shed greater light on the flame NO problem. These investigators measured the prompt NO formed as a function of equivalence ratio for many hydrocarbon compounds. Their results are shown in Fig. 8.4. What is significant about these results is that the maximum prompt NO is reached on the fuel-rich side of stoichiometric, remains at a high level through a fuel-rich region, and then drops off sharply at an equivalence ratio of about 1.4. Bachmeier et al. also measured the HCN concentrations through propane– air flames. These results, which are shown in Fig. 8.5, show that HCN

426

Combustion

ppmNO ppmNO

80

60 60 40 40 20

i-Octane

Gasoline

20

0 0

ppmNO 40

ppmNO

60

20

40 n-Hexazne

0

20

Cyclohexane

ppmNO 0 100

40 20

ppmNO

n-Butane

50

0

Benzene

ppmNO 0

40

ppmNO

20

200

Propane 0 40

100

ppmNO

Acetylene

0 ppmNO

20 0 40

Ethane

60

ppmNO

20

40

0 0.6

Ethylene

20

Methane

0 1.0 1.4 Equivalence ratio

0.8

1.2 1.6 Equivalence ratio

2.0

FIGURE 8.4 Prompt NO as a function of mixture strength and fuel. The dotted lines show the uncertainty of the extrapolation at the determination of prompt NO in propane flames; similar curves were obtained for the other hydrocarbons (from Bachmeier et al. [13]).

25 φ  1.5

ppmHCN

20 15 10 φ  1.35

5 0

φ  1.1 5

0

10

15

20

Time (ms) 0 FIGURE 8.5

10

20

30 40 Height

50 mm

HCN profiles of fuel-rich propane–air flames (from Bachmeier et al. [13]).

Environmental Combustion Considerations

427

concentrations rise sharply somewhere in the flame, reach a maximum, and then decrease sharply. However, for an equivalence ratio of 1.5, a fuel-rich condition for which little prompt NO is found, the HCN continues to rise and is not depleted. The explanation offered for this trend is that HCN forms in all the rich hydrocarbon flames; however, below an equivalence ratio of 1.4, O radicals are present in sufficient abundance to deplete HCN and form the NO. Since the sampling and analysis techniques used by Bachmeier et al. [13] did not permit the identification of the cyanogen radical CN, the HCN concentrations found most likely represent the sum of CN and HCN as they exist in the flame. The CN and HCN in the flame are related through the rapid equilibrium reactions [14] CN  H 2  HCN  H

(8.57)

CN  H 2 O  HCN  OH

(8.58)

The HCN concentration is probably reduced mainly by the oxidation of the CN radicals [14,15]. From other more recent studies of NO formation in the combustion of lean and slightly rich methane–oxygen–nitrogen mixtures as well as lean and very rich hydrocarbon–oxygen–nitrogen mixtures, it must be concluded that some of the prompt NO is due to the overshoot of O and OH radicals above their equilibrium values, as the Bowman and Seery results suggested. But even though O radical overshoot is found on the fuel-rich side of stoichiometric, this overshoot cannot explain the prompt NO formation in fuel-rich systems. It would appear that both the Zeldovich and Fenimore mechanisms are feasible. Some very interesting experiments by Eberius and Just [16] seem to clarify what is happening in the flame zone with regard to NO formation. Eberius and Just’s experiments were performed on a flat flame burner with propane as the fuel. Measurements were made of the prompt NO at various fuel–oxygen equivalence ratios whose flame temperatures were controlled by dilution with nitrogen. Thus a range of temperatures could be obtained for a given propane– oxygen equivalence ratio. The results obtained are shown in Fig. 8.6. The highest temperature point for each equivalence ratio corresponds to zero dilution. The shapes of the plots in Fig. 8.6 are revealing. At both the low- and hightemperature ends, all the plots seem nearly parallel. The slopes at the lowtemperature end are very much less than the slopes at the high-temperature end, thereby indicating two mechanisms for the formation of prompt NO. The two mechanisms are not related solely to the fuel-rich and fuel-lean stoichiometry, as many investigators thought, but also to the flame temperature. These results suggest two routes–a high-temperature, high-activation route and a lower-temperature, low-activation route. The systematic appearance of these data led Eberius and Just to estimate the activation energy for the two regions. Without correcting for diffusion, they

428

Combustion

120 100

NO (ppm)

80

φ  1.4 φ  1.3 φ  1.2 φ  1.1 φ  1.0 φ  0.9

60 40 20 0 1700

1900

2100 T (K)

2300

2500

FIGURE 8.6 Prompt NO as a function of the temperature at various mixture strengths φ in adiabatic propane–synthetic air flames (from Eberius and Just [16]).

obtained an activation energy of the order of 270 kJ/mol for the high-temperature zone. This value is remarkably close to the 315 kJ/mol activation energy required for the initiating step in the Zeldovich mechanism. Furthermore, diffusion corrections would raise the experimental value somewhat. The lowtemperature region has an activation energy of the order of 50–60 kJ/mol. As will be shown later, radical attack on the cyano species is faster than oxygen radical attack on hydrogen. The activation energy of O  H2 is about 33 kJ/mol; therefore, the HCN reaction should be less. Again, diffusion corrections for the oxygen atom concentration could lower the apparent activities of activation energies of 50–60 kJ/mol to below 34 kJ/mol. This crude estimate of the activation energy from Eberius and Just’s low-temperature region, together with the formation of HCN found by the same group [13] in their other flame studies with propane (Fig. 8.5), appears to indicate that the Fenimore mechanism [9] would hold in the lower-temperature region. The kinetic details for prompt NO formation must begin with the possible reactions between N2 and hydrocarbon fragments, as Fenimore [9] originally suggested. Hayhurst and Vance [17] suggest that two other likely candidate reactions may be added to those posited by Fenimore. The four candidate reactions would then be C2  N 2 → CN  CN

(8.59)

C2 H  N 2 → HCN  CN

(8.60)

CH  N 2 → HCN  N

(8.61)

CH 2  N 2 → HCN  NH

(8.62)

Environmental Combustion Considerations

429

As discussed in the introduction to Chapter 4, the existence of C2 and CH in hydrocarbon–air flames is well established. Methylene (CH2) arises in most combustion systems by OH and H attack on the methyl radical (CH3). Similar attack on CH2 creates CH. Hayhurst and Vance [17] established that the amount of prompt NO in moderately fuel-rich systems is proportional to the number of carbon atoms present per unit volume and is independent of the original parent hydrocarbon identity. This result indicates that reactions (8.59) and (8.60) are not primary contributors because it is unlikely that C2 or C2H could derive from CH4 with an efficiency one-half that from C2H2 or one-third that from C3H6 or C3H8. In their complete model Miller and Bowman [6] introduced the reactions CH 2 N 2  H 2 CN  N

(8.63)

C  N 2  CN  N

(8.64)

and

They conclude from estimated rates that reaction (8.63) is an insignificant contributor to prompt NO. However, they also point out that the reverse reaction (8.64) is very fast at room temperature and under shock tube conditions. Hence this step is a minor, but nonnegligible contributor to prompt NO and, because of the large endothermicity of reaction (8.64), its importance with respect to reaction (8.61) increases with increasing temperature. The major products of reactions (8.61) and (8.62) are HCN and NH. It is to be noted that, experimentally, HCN and CN are indistinguishable in hydrocarbon flames because of the equilibrium reactions (8.57) and (8.58) mentioned with respect to the work of Bachmeier et al. [13]. In consideration of the work of Ref. [1], the major kinetic route to prompt NO would then appear to be HCN  O  NCO  H

(8.65)

NCO  H  NH  CO

(8.66)

NH  (H, OH)  N  (H 2 , H 2 O)

(8.67)

N  OH  NO  H

(8.68)

N  O2  NO  O

(8.69)

Following the conclusions of Bowman [1], then, from the definition of prompt NO, these sources of prompt NO in hydrocarbon fuel combustion can be identified: (1) nonequilibrium O and OH concentrations in the reaction zone and burned gas, which accelerate the rate of the thermal NO mechanism; (2) a reaction sequence, shown in Fig. 8.7, that is initiated by reactions of hydrocarbon radicals, present in and near the reaction zone, with molecular nitrogen

430

Combustion

C H, OH H2 CHx

H2O

N2 HCN

O

OH

O

O2 OH NCO

O

Fuel-N

CN

H OH H

NH H OH

H2 HNCO

N

H

NO N2 OH O2 NO

NH2

CHx (x 1,2,3) FIGURE 8.7 Reaction path diagram illustrating the major steps in prompt NO formation, the conversion of fuel nitrogen in flames, and reburning (from Bowman [12]).

(the Fenimore prompt NO mechanism); and (3) reaction of O atoms with N2 to form N2O via the three-body recombination reaction, O  N2  M → N2 O  M and the subsequent reaction of the N2O to form NO via N 2 O  O → NO  NO The relative importance of these three mechanisms in NO formation and the total amount of prompt NO formed depend on conditions in the combustor. Acceleration of NO formation by nonequilibrium radical concentrations appears to be more important in non-premixed flames, in stirred reactors for lean conditions, and in low-pressure premixed flames, accounting for up to 80% of the total NO formation. Prompt NO formation by the hydrocarbon radical–molecular nitrogen mechanism is dominant in fuel-rich premixed hydrocarbon combustion and in hydrocarbon diffusion flames, accounting for greater than 50% of the total NO formation. Nitric oxide formation by the N2O mechanism increases in importance as the fuel–air ratio decreases, as the burned gas temperature decreases, or as pressure increases. The N2O mechanism is most important under conditions where the total NO formation rate is relatively low [1]. Bozzelli and Dean (17a) have recently proposed that NO can also form through the formation of NNH radicals and their subsequent oxidation via the reaction sequence N 2  H  NNH NNH  O  NO   Experimental evidence for this mechanism has been found in low pressure H2/ air flames, fuel-rich H2/O2/N2 and CH4/O2/N2 flames at atmospheric pressure and

431

Environmental Combustion Considerations

NO mol fraction (ppm)

103

0.5% Added pyridine 102 Base oil

101

0.6

0.8

1.0

1.2

φ FIGURE 8.8 Nitric oxide emissions from an oil-fired laboratory furnace (from Martin and Berkau [18]).

in lean H2/air mixtures in stirred reactors. Modeling of these systems by Konnov et al. (17b) supports the importance of this mechanism. Through modeling and experimental studies of jet stirred reactors fuels with various hydrocarbons, Ruter et al. (17c) found prompt and NNH pathways dominate the formation of NO in the flame, while Zeldovich and nitrous oxide are primary contributors to the NO formation in the post flame. While some uncertainty exists in the NNH rate constants, the NNH mechanism appears to play a role in NO formation. Fuel-bound nitrogen NO mechanisms: In several recent experiments, it has been shown that NO emissions from combustion devices that operate with nitrogen-containing compounds in the fuel are high; in other words, fuel-bound nitrogen is an important source of NO. The initial experiments of Martin and Berkau [18] commanded the greatest interest. These investigators added 0.5% pyridine to base oil and found almost an order of magnitude increase over the NO formed from base oil alone. Their results are shown in Fig. 8.8. During the combustion of fuels containing bound nitrogen compounds, the nitrogen compounds most likely undergo some thermal decomposition prior to entering the combustion zone. Hence, the precursors to NO formation will, in general, be low-molecular-weight, nitrogen-containing compounds or radicals (NH3, NH2, NH, HCN, CN, etc.). All indications are that the oxidation of fuelbound nitrogen compounds to NO is rapid and occurs on a time scale comparable to the energy-release reactions in the combustion systems. This conclusion arises from the fact that the NH and CN oxidation reactions discussed in the previous section are faster [19,20] than the important chain branching reaction O  H 2 → OH  H

(8.70)

Thus, the reaction system cannot be quenched to prevent NO formation from fuel-bound nitrogen, as is the case with atmospheric nitrogen. In fact, in the

432

Combustion

vicinity of the combustion zone, observed NO concentrations significantly exceed calculated equilibrium values. In the post-combustion zone, the NO concentration decreases relatively slowly for fuel-lean mixtures and more rapidly for fuel-rich mixtures. Recall Bowman and Seery’s results (Fig. 8.3) showing that fuel-rich systems approach equilibrium faster. When fuel–nitrogen compounds are present, high NO yields are obtained for lean and stoichiometric mixtures and relatively lower yields are found for fuel-rich mixtures. The NO yields appear to be only slightly dependent on temperature, thus indicating a low activation energy step. This result should be compared to the strong temperature dependence of NO formation from atmospheric nitrogen. The high yields on the lean side of stoichiometric pose a dilemma. It is desirable to operate lean to reduce hydrocarbon and carbon monoxide emissions; but with fuel containing bound nitrogen, high NO yields would be obtained. The reason for the superequilibrium yields is that the reactions leading to the reduction of NO to its equilibrium concentration, namely, O  NO → N  O2

(8.71)

NO  NO → N 2 O  O

(8.72)

NO  RH → products

(8.73)

are very slow. NO can be reduced under certain conditions by CH and NH radicals, which can be present in relatively large concentrations in fuel-rich systems. These reduction steps and their application will be discussed later. The extent of conversion of fuel nitrogen to NO is nearly independent of the parent fuel molecule, but is strongly dependent on the local combustion environment and on the initial fuel nitrogen in the reactant. Unlike sulfur in the fuel molecule, nitrogen is much more tightly bound in the molecule and, for the most part, in an aromatic ring [21]. Regardless, all fuel–nitrogen compounds exhibit solely carbon–nitrogen or nitrogen–hydrogen bonding. Thus, it is not surprising that in the oxidation of fuel–nitrogen compounds, the major intermediates are HCN and CN and amine radicals stemming from an ammonia structure, that is, NH2, NH, and N. In a large radical pool, there exists an equilibrium, NHi  X  NHi1  XH

(8.74)

which essentially establishes an equilibrium between all NH compounds, that is, NH3  NH 2  NH  N

(8.75)–(8.77)

Consequently, reactions (8.61) and (8.62) can be written as a generalized reaction CHi  N 2 → HCN  NHi1

(8.78)

Thus, there is great similarity between the prompt NO reactions discussed and the fuel–nitrogen reactions.

433

Environmental Combustion Considerations

In the combustion of fuel–nitrogen compounds, the equilibrium HCN  H  CN  H 2

(8.57)

will certainly exist. Thus, the conversion of all relevant intermediates to NO has essentially been discussed, and only the NH2 reactions remain to be considered. These reactions follow the sequence NH 2  H → NH  H 2

(8.79)

NH 2  OH → NH  H 2 O

(8.80)

NH 

{HOH} → N  ⎪⎧⎪⎨⎪⎪⎩HH O⎪⎫⎪⎬⎪⎪⎭ 2

(8.81)

2

The general scheme of the fuel–nitrogen reactions is also represented in Fig. 8.7. In fuel-rich systems, there is evidence [8, 22, 23] that the fuel–nitrogen intermediate reacts not only with oxidizing species in the manner represented, but also competitively with NO (or another nitrogen intermediate) to form N2. This second step, of course, is the reason that NO yields are lower in fuel-rich systems. The fraction of fuel nitrogen converted to NO in fuel-rich systems can be as much as an order of magnitude less than that of lean or near-stoichiometric systems. One should realize, however, that even in fuel-rich systems, the exhaust NO concentration is substantially greater than its equilibrium value at the combustion temperature. Haynes et al. [14] have shown that when small amounts of pyridine are added to a premixed, rich (φ  1.68; T  2030 K) ethylene–air flame, the amount of NO increases with little decay of NO in the post-flame gases. However, when larger amounts of pyridine are added, significant decay of NO is observed after the reaction zone. When increasingly higher amounts of pyridine are added, high concentrations of NO leave the reaction zone, but this concentration drops appreciably in the post-flame gases to a value characteristic of the flame, but well above the calculated equilibrium value. Actual experimental results are shown in Fig. 8.9. In fuel-rich systems, the conversion reactions of the fuel–nitrogen intermediates are subject to doubt, mainly because the normal oxidizing species O2, O, and OH are present only in very small concentrations, particularly near the end of the reaction zone. Haynes et al. [14] offer the interesting suggestion that the CN can be oxidized by CO2 since the reaction CN  CO2 → OCN  CO

(8.82)

is 85 kJ/mol exothermic and estimated to be reasonably fast.

b. Nitrogen Dioxide Reaction Mechanisms Significant concentrations of NO2 have been reported in the exhaust of gas turbines and in the products of range-top burners [21]. These results are surprising

434

Combustion

80

NO concentration (ppm)

C 60

Increasing pyridine addition

40

B D

20 A 0

0

4

Equil

8

12

Time (ms) FIGURE 8.9 Effect of NO concentrations leaving the reaction zones of an ethylene–air flame (φ  1.68, T  2030 K) with various pydrine additions. Curve A, no pyridine addition; curves B and C, 0.1–0.5 N by weight of fuel; and curve D, NO addition to the fuel–air mixture (from Haynes et al. [14]).

because chemical equilibrium considerations reveal that the NO2/NO ratio should be negligibly small for typical flame temperatures. Furthermore, when kinetic models are modified to include NO2 formation and reduction, they show that the conversion of NO to NO2 can be neglected in practical devices. However, in the case of sampling from gas turbines, NO2 can vary from 15% to 50% of the total NOx, depending on the NO level [21, 24, 25]. In the case of range-type burners, the NOx has been reported as high as 15–20 times the NO levels in parts of the flame surrounding the burner top [26, 27]. Merryman and Levy [28] examined both NO and NO2 formation in a flat flame burner operated near stoichiometric. In the low-temperature regime of visible flames, they found large concentrations of HO2 that can react with the NO formed in the high-temperature regime and diffuse back to the lowertemperature zone. Their measurements showed that NO2 is produced in the visible regime of all air flames (with and without fuel-bound nitrogen) and that NO is observed only in the visible region when fuel-bound nitrogen is present. Furthermore, these investigators found that NO2 is consumed rapidly in the near-post-flame zone, whereupon the NO concentration rises correspondingly. They postulated the following scheme to represent their findings:

}

HN  O2 → NO  CN

(8.83)

NO  HO2 → NO2  OH

(8.84)

NO2  O → NO  O2

(8.85)

Environmental Combustion Considerations

435

In light of Fig. 8.7, whether reaction (8.83) should be the representative reaction for NO formation is irrelevant. The significant step is represented by reaction (8.84). One should recall that there can be appreciable amounts of HO2 in the early parts of a flame. The appearance of the NO2 is supported further by the fact that reaction (8.84) is two orders of magnitude faster than reaction (8.85). The importance of the hydroperoxy radical attack on NO appeared to be verified by the addition of NO to the cold-fuel mixtures in some experiments. In these tests, the NO disappeared before the visible region was reached in oxygen-rich and stoichiometric flames, that is, flames that would produce HO2. The NO2 persists because, as mentioned previously, its reduction to N2 and O2 is very slow. The role of HO2 would not normally be observed in shock tube experiments owing to the high temperatures at which they usually operate. The Merryman–Levy sequence could explain the experimental results that show high NO2/NO ratios. For the experiments in which these high ratios were found, it is quite possible that reaction (8.85) is quenched, in which case the NO2 is not reduced. Cernansky and Sawyer [29], in experiments with turbulent diffusion flames, also concluded that the high levels of NO2 found were due to the reactions of NO with HO2 and O atoms. The experimental efforts reporting high NO2 levels have come into question because of the possibility that much of the NO2 actually forms in sampling tubes [30, 31]. Optical techniques are now being applied; but, unfortunately, the low concentrations of NO2 that exist make resolution of the controversy very difficult.

c. Nitrous Oxide Formation Mechanisms Quoting directly from Bowman [1], The principal gas-phase reactions forming N2O in fossil fuel combustion are NCO  NO → N 2 O  CO NH  NO → N 2 O  H In natural gas combustion an increasingly important contribution from O  N2  M → N2O  M occurs in fuel-lean mixtures and at low temperature and elevated pressures. The primary N2O removal steps are H  N 2 O → N 2  OH and O  N 2 O → N 2  O2 → NO  NO Calculated lifetimes of N2O in combustion products indicate that for temperatures above 1500 K, the lifetime of N2O typically is less than 10 ms, suggesting that except for low-temperature combustion, as found in fluidized bed combustors and in some postcombustion NO removal systems, N2O emissions should not be significant, a conclusion that is in agreement with the most recent measurements of N2O emissions from combustion devices.

436

Combustion

4. The Reduction of NOx Because of the stringent emissions standards imposed on both mobile and stationary power sources, methods for reducing NOx must be found; moreover, such methods should not impair the efficiency of the device. The simplest method of reducing NOx, particularly from gas turbines, is by adding water to the combustor can. Water vapor can reduce the O radical concentration by the following scavenging reaction: H 2 O  O → 2OH

(8.86)

Fortunately, OH radicals do not attack N2 efficiently. However, it is more likely that the effect of water on NOx emissions is through the attendant reduction in combustion temperature. NOx formation from atmospheric nitrogen arises primarily from the very temperature-sensitive Zeldovich mechanism. The problem of NOx reduction is more difficult in heterogeneous systems such as those which arise from direct liquid fuel injection and which are known to burn as diffusion flames. One possible means is to decrease the average droplet size formed from injection. Kesten [32] and Bracco [33] have shown that the amount of NO formed from droplet diffusion flames can be related to the droplet size; viz., one large droplet will give more NO than can be obtained from a group of smaller droplets whose mass is equal to that of the larger droplet. Any means of decreasing the heterogeneity of a flame system will decrease the NOx. Another possible practical scheme is to emulsify the fuel with a higher vapor pressure, nonsoluble component such as water. It has been shown [34] that droplets from such emulsified fuels explode after combustion has been initiated. These microexplosions occur when the superheated water within the fuel droplet vaporizes, hence appreciably decreasing the heterogeneity of the system. A further benefit is obtained not only because the water is available for dilution, but also because the water is present in the immediate vicinity of the diffusion flame. If it is impossible to reduce the amount of NOx in the combustion section of a device, the NOx must be removed somewhere in the exhaust. Myerson [35] has shown that it is possible to reduce NOx by adding small concentrations of fuel and oxygen. The addition of about 0.1% hydrocarbon (isobutane) and 0.4% O2 to a NOx-containing system at 1260 K reduced the NOx concentration by a factor of 2 in about 125 ms. Myerson [35] found that the ratio of O2/HC was most important. When the concentrations of O2 and the hydrocarbon are large, a HCN-formation problem could arise. This procedure is feasible only for slightly fuel-lean or fuel-rich systems. The oxygen is the creator and the destroyer of other species involved in the NO reduction. This fact, in turn, means that the initial addition of O2 to the hydrocarbon–NO mixture promotes the production of the strongly reducing species CH and CH2 and similar substituted free radicals that otherwise must be produced by slower pyrolysis reactions.

437

Environmental Combustion Considerations

Continued addition of O2 beyond one-half the stoichiometric value with the hydrocarbons present encourages a net destruction of the hydrocarbon radicals. For the temperature range 1200–1300 K, production of the hydrocarbon radicals via hydrogen abstraction by O2 is rapid, even assuming an activation energy of 520 kJ/mol, and more than adequate to provide sufficient radicals for NO reduction in the stay time range of 125 ms. Myerson postulated that the following reactions are involved: CH  NO → HCO  N  217 kJ

(8.87)

CH  NO → HCN  O  305 kJ

(8.88)

The exothermicity of reaction (8.87) is sufficient to fragment the formyl radical and could be written as CH  NO → H  CO  N  104 kJ

(8.89)

In the absence of O2, the N radicals in these fuel-rich systems can react rapidly with NO via N  NO → N 2  O  305 kJ

(8.90)

Another technique currently in practice is known as the thermal DeNOx, process [36], which uses ammonia as the NOx reduction agent. The ammonia is injected into the exhaust gases of stationary power plants burning fossil fuels. The process is effective in a narrow temperature range, about T  1250 K. Below about 1100 K, the reaction takes place too slowly to be of value, and about 1400 K more NO is formed. Miller et al. [37] discovered that if H2 is added to the system, the center of the temperature window moves to a lower value without changing the width of the window. They also found that slightly lean combustion products appear to be required for the reduction reaction to be effective; that is, the process is implemented under excess oxygen conditions. Increasing the NH3 concentration to a comparable O2 concentration inhibits the process under certain conditions, and the presence of water slightly inhibits the NO reduction because the optimum temperature is increased slightly [6]. Explaining these effects has been one of the successes of kinetic modeling [6]. The ammonia in the process is considered to form the amine radical NH2, which reacts with the NO to form an intermediate that decays into products: H NH2  NO

N

N

O

products

(8.91)

H

There are various possibilities as to the fate of the intermediate. It could form HNNO  H. However, this route is unlikely because no H atoms are

438

Combustion

found at low temperatures; nor is there any N2O, which would inevitably have to form. In addition, this decay step is endothermic. Another possibility is the formation of N2O and H2, which is exothermic. But there is a large energy barrier involved in forming H2. Also, of course, no N2O is found. The formation of H2N¨N and O is very endothermic and is not conceivable. The possibility exists of an migration of an H atom. Because the migration step H

H N

N

N

N

O

H

O

N

N H

H

O

(8.92)

H

is also very endothermic, it is ruled out. But what appears to be most feasible is the migration of H to the O atom in the following step: H

H N

N

N

O

HNN  OH

N

(8.93)

OH

H

The product HNN provides a plausible route for the overall NO reduction mechanism, which permits the determination of the temperature window. Miller and Bowman [6] proposed the competitive channels shown in Table 8.2 as the explanation.

TABLE 8.2 NO Reduction Scheme NO Reduction Scheme NH3 (OH, O) (NO)

NH2 (H2O, OH)

(OH)

(NO) NNH (OH)

NH (H2O)

(NO) N2  HNO (NH2) NO  NH3

(OH) (M)

NO  H (M) NO (H2O) Channel 1

(O2, OH) HNO (O, H, NH) (M) NO  H (M)

(OH) NO (H2O)

Channel 2

Environmental Combustion Considerations

439

The thermal DeNOx system removes NO in practical systems because the NH2  NO initiates a significant chain branching system, thereby allowing the overall reaction sequence to be self-sustaining. Following the general scheme in Table 8.2, the conversion of NH3 to NH2 occurs principally by reaction with OH: NH3  OH  NH 2  H 2 O But, in the absence of water vapor, it may occur by reaction with O atoms: NH3  O  NH 2  OH The required chain branching to regenerate OH and O and hence to continue the conversion is accomplished [6] by the reaction sequence for Channel 1: NH 2  NO  NNH  OH NNH  NO  N 2  HNO HNO    H    M The H atom produced in the last step reacts with O2 in the familiar chain branching step H  O2  OH  O giving an overall chain branching radical pool. In the presence of the water inherent in the product composition of most combustion systems, the system is further augmented by the reaction of O atoms with water via O  2 O  OH  OH and OH becomes the dominant species converting NH3 to NH2. In their model, Miller and Bowman [6] consider the important chaintermination steps to be NH 2  NO  N 2  H 2 O NH 2  HNO  NH3  NO and OH  O  H 2 O  NO They conclude that, at the low-temperature end of the effective temperature window, the NO reduction effectiveness is limited principally by the rates of the chain-termination reactions that compete with the preceding branching sequence. In addition, below about 1100 K, hydrogen abstraction by OH is so

440

Combustion

slow that little NH2 forms and H  O2  M → HO2  M becomes a competitive step for H  O2 → OH  O. In the temperature range 1100–1400 K, the mix of branching and termination is proper for the conversion of NH3 to NH2 at a sufficiently rapid and sustaining rate. However, above about 1400 K, Channel 2 becomes dominant. At such temperatures, there is more chain branching, which leads to a higher concentration of OH. Thus, the Channel 2 reaction NH 2  OH  NH  H 2 O is favored over the Channel 1 reaction NH 2  NO → NNH  OH These two reactions are competitive around 1250 K. Note that in Channel 1, two NO molecules react to form one N2 and one NO so that an overall reduction in NO is obtained. The role of added H2 manifests itself by increasing the amount of chain branching at the lower temperatures and increasing the overall concentration of OH. Thus the window shifts to lower temperatures. Other post-combustion NOx removal techniques include the injection of urea ([NH2]2CO) and cyanuric acid ([HOCN]3). The latter is termed the RAPRENO process [37a]. When heated, cyanuric acid sublimes and decomposes to form isocyanic acid HNCO. Similarly, urea reacts to form NH3 and HNCO. Thus its NOx reduction path follows that of the thermal DeNOx route as well as that of the RAPRENO route to be discussed. The major route in the RAPRENO process is the radical attack on the isocyanuric acid by H and OH via H  HNCO  NH 2  CO OH  HNCO  NCO  H 2 O The first of these two steps forms the amine radical NH2 and it acts as in the thermal DeNOx process. The importance of the CO is that its oxidation produces H atoms from the well known step CO  OH  CO2  H The importance of the second step is that it provides the primary NO removal step [6] NCO  NO  N 2 O  CO for the process. The N2O decays, as discussed earlier, to form N2. The advantage of the RAPRENO process may be its ability to remove NOx at much lower temperatures than the thermal DeNOx process. There is some indication that this lower-temperature effect may be to decomposition of cyanuric acid on surfaces.

Environmental Combustion Considerations

441

Considering that wet scrubbers are in place in many facilities and more are planned for the future, another efficient means for NOx removal could be considered. These scrubbing methods are limited by the relatively inert nature of NO. It has been proposed that this difficulty can be overcome by the conversion of NO to the much more active NO2 through reaction (8.84), as discussed earlier: NO  HO2 → NO2  OH The question arises as to how to produce HO2 radicals. For post-combustion conversion, the obvious candidate would be an aqueous solution of hydrogen peroxide H2O2. Although hydrogen peroxide readily converts to water and oxygen through a heterogeneous decomposition, its homogeneous decomposition route follows the simple steps H 2 O2  M → 2OH  M OH  H 2 O2 → H 2 O  HO2 to form the necessary hydroperoxy radical HO2. However, one must realize that this simple reaction sequence is so slow that it is ineffective below 600 K. At high temperatures, particularly at atmospheric pressure, HO2 dissociates. Moreover, it would appear that at temperatures above 1100 K, the general radical pool is large enough that recombination reactions become too competitive. Thus, this aqueous process has an effective temperature window between 600 and 1100 K [6].

D. SOx EMISSIONS Sulfur compounds pose a dual problem. Not only do their combustion products contribute to atmospheric pollution, but these products are also so corrosive that they cause severe problems in the operation of gas turbines and industrial power plants. Sulfur pollution and corrosion were recognized as problems long before the nitrogen oxides were known to affect the atmosphere. For a time, the general availability of low-sulfur fuels somewhat diminished the general concern with respect to the sulfur. However, the possibility that China is developing its huge coal resources has again raised the specter of massive sulfur oxide emissions. Sulfur may be removed from residual oils by catalytic hydrodesulfurization techniques, but the costs of this process are high and the desulfurized residual oils have a tendency to become “waxy” at low temperatures. To remove sulfur from coal is an even more imposing problem. It is possible to remove pyrites from coal, but this approach is limited by the size of the pyrite particles.

442

Combustion

Unfortunately, pyrite sulfur makes up only half the sulfur content of coal, while the other half is organically bound. Coal gasification is the only means by which this sulfur mode can be removed. Of course, it is always possible to eliminate the deleterious effects of sulfur by removing the major product oxide SO2 by absorption processes. These processes impose large initial capital investments. The presence of sulfur compounds in the combustion process can affect the nitrogen oxides, as well. Thus, it is important to study sulfur compound oxidation not only to find alternative or new means of controlling the emission of objectionable sulfur oxides, but also to understand their effect on the formation and concentration of other pollutants, especially NOx. There are some very basic differences between the sulfur problem and that posed by the formation of the nitrogen oxides. Nitrogen in any combustion process can be either atmospheric or organically bound. Sulfur can be present in elemental form or organically bound, or it may be present as a species in various inorganic compounds. Once it enters the combustion process, sulfur is very reactive with oxidizing species and, in analogy with fuel nitrogen, its conversion to the sulfurous oxides is fast compared to the other energy-releasing reactions. Although sulfur oxides were recognized as a problem in combustion processes well before the concern for photochemical smog and the role of the nitrogen oxides in creating this smog, much less is understood about the mechanisms of sulfur oxidation. Indeed, the amount of recent work on sulfur oxidation has been minimal. The status of the field has been reviewed by Levy et al. [38] and Cullis and Mulcahy [39] and much of the material from the following subsections has been drawn from Cullis and Mulcahy’s article.

1. The Product Composition and Structure of Sulfur Compounds When elemental sulfur or a sulfur-bearing compound is present in any combustion system, the predominant product is sulfur dioxide. The concentration of sulfur trioxide found in combustion systems is most interesting. Even under very lean conditions, the amount of sulfur trioxide formed is only a few percent of that of sulfur dioxide. Generally, however, the sulfur trioxide concentration is higher than its equilibrium value, as would be expected from the relation SO2  21 O2  SO3

(8.94)

These higher-than-equilibrium concentrations may be attributable to the fact that the homogeneous reactions that would reduce the SO3 to SO2 and O2 are slow. This point will be discussed later in this section. It is well known that SO3 has a great affinity for water and that at low temperatures it appears as sulfuric acid H2SO4. Above 500°C, sulfuric acid dissociates almost completely into sulfur trioxide and water.

443

Environmental Combustion Considerations

Under fuel-rich combustion conditions, in addition to sulfur dioxide, the stable sulfur products are found to be hydrogen sulfide, carbonyl sulfide, and elemental sulfur. Owing to their reactivity, other oxides of sulfur may appear only as intermediates in various oxidation reactions. These are sulfur monoxide SO, its dimer (SO)2, and disulfur monoxide S2O. Some confusion has attended the identification of these oxides; for example, what is now known to be S2O was once thought to be SO or (SO)2. The most important of these oxides is sulfur monoxide, which is the crucial intermediate in all high-temperature systems. SO is a highly reactive radical whose ground state is a triplet which is electronically analogous to O2. According to Cullis and Mulcahy [39], its lifetime is seldom longer than a few milliseconds. Spectroscopic studies have revealed other species in flames, such as CS, a singlet molecule analogous to CO and much more reactive; S2, a triplet analogous to O2 and the main constituent of sulfur vapor above 600°C; and the radical HS. Johnson et al. [40] calculated the equilibrium concentration of the various sulfur species for the equivalent of 1% SO, in propane–air flames. Their results, as a function of fuel–air ratio, are shown in Fig. 8.10. The dominance of SO2 in the product composition for these equilibrium calculations, even under deficient air conditions, should be noted. As reported earlier, practical systems reveal SO3 concentrations that are higher (1–2%) than those depicted in Fig. 8.10.

Mole percentage of sulphur species

100

SO2

10

SO

S

1

SH 0.1 COS SO3

S2

0.01

CS 0.001 60

H2S 80

100

120

Fuel-air ratio (% stoich air) 1867 2141 2240 2071 Adiabatic temperature (K) FIGURE 8.10 Equilibrium distribution of sulfur-containing species in propane–air flames with unburned gases initially containing 1% SO2 (from Johnson et al. [40]).

444

Combustion

TABLE 8.3 Structure of Gaseous Sulfur Compounds Sulfur8

Rhombic

Sulfur monoxide SO

S

Sulfur dioxide SO2 (OSO)

O O

O S

118°

S

O

O

Sulfur superoxide SO2 S (SOO)

O

O

Sulfur trioxide SO3

O

O O

O

S

S

120°

O

O Sulfur suboxide S2O

S

S

O

Carbonyl sulfide COS

S

C

O

S

C

O

Carbon disulfide CS2

S

C

S

S

C

S

Organic thiols

R

SH

Organic sulfides

R

S

R

Organic disulfides

R

S

S

S

C

O

R

Insight into much that has and will be discussed can be obtained by the study of the structures of the various sulfur compounds given in Table 8.3.

2. Oxidative Mechanisms of Sulfur Fuels Sulfur fuels characteristically burn with flames that are pale blue, sometimes very intensely so. This color comes about from emissions as a result of the reaction O  SO → SO2  hv (8.95) and, since it is found in all sulfur–fuel flames, this blue color serves to identify SO as an important reaction intermediate in all cases. Most studies of sulfur–fuel oxidation have been performed using hydrogen sulfide, H2S, as the fuel. Consequently, the following material will concentrate on understanding the H2S oxidation mechanism. Much of what is learned from

445

Environmental Combustion Considerations

240

200

Pressure (torr)

160

Explosion

120

80

40

No explosion

Ignition 0 220 FIGURE 8.11 and oxygen.

260 300 Temperature (°C)

340

Approximate explosion limits for stoichiometric mixtures of hydrogen sulfide

this mechanism can be applied to understanding the combustion of COS and CS2 as well as elemental and organically bound sulfur.

a. H2S Figure 8.11 is a general representation of the explosion limits of H2S/O2 mixtures. This three-limit curve is very similar to that shown for H2/O2 mixtures. However, there is an important difference in the character of the experimental data that determine the H2S/O2 limits. In the H2S/O2 peninsula and in the third limit region, explosion occurs after an induction period of several seconds. The main reaction scheme for the low-temperature oxidation of H2S, although not known explicitly, would appear to be H 2 S  O2 → SH  HO2

+ 176 kJ

(8.96)

SH  O2 → SO  OH

88 kJ

(8.97)

H 2 S  SO → S2 O  H 2

29 kJ

(8.98)

OH  H 2 S → H 2 O  HS

125 kJ

(8.99)

446

Combustion

The addition of reaction (8.98) to this scheme is necessary because of the identification of S2O in explosion limit studies. More importantly, Merryman and Levy [41] in burner studies showed that S2O occurs upstream from the peak of the SO concentration and that elemental sulfur is present still further upstream in the preignition zone. The most probable system for the introduction of elemental sulfur is SH  SH → H 2 S  S S  SH → S2  H

13 kJ

(8.100)

63 kJ

(8.101)

Given the preignition zone temperatures and overall pressures at which the flame studies were carried out, it does not seem kinetically feasible that the reaction S2  O  M → S2 O  M (8.102) could account for the presence of S2O. The disproportionation of SO would have to give SO2 as well as S2O. Since SO2 is not found in certain experiments where S2O can be identified, disproportionation would not be feasible, so reaction (8.98) appears to be the best candidate for explaining the presence of S2O. Reaction (8.97) is the branching step in the mechanism. It has been suggested that SH  O2  M → HSO2  M (8.103) competes with reaction (8.97), thus determining the second limit. Cullis and Mulcahy [39] suggested the reaction S2 O  O2 → SO2  SO

(8.104)

as the degenerate branching step. The explicit mechanism for forming S2O and its role in flame processes must be considered an uncertainty. At higher temperatures, the reaction O2  SO → O  SO2

(8.105)

becomes competitive with reaction (8.98) and introduces O radicals into the system. The presence of O radicals gives another branching reaction, namely, O  H 2 S → OH  SH

(8.106)

The branching is held in check by reaction (8.98), which removes SO, and the fast termolecular reaction O  SO  M → SO2  M which removes both O radicals and SO.

(8.107)

447

Environmental Combustion Considerations

In shock tube studies, SO2 is formed before OH radicals appear. To explain this result, it has been postulated that the reaction O  H 2 S → SO  H 2

+ 222 kJ

(8.108)

is possible. This reaction and reaction (8.98) give the overall step O2  H 2 S → SO2  H 2

(8.109)

Detailed sampling in flames by Sachjan et al. [42] indicates that the H2S is oxidized in a three-step process. During the first stage, most of the H2S is consumed, and the products are mainly sulfur monoxide and water. In the second stage, the concentration of SO decreases, the concentration of OH reaches its maximum value, the SO2 reaches its final concentration, and the concentration of the water begins to build as the hydrogen passes through a maximum. The interpretation given to these results is that, during the first stage, the H2S and O2 are consumed mainly by reactions (8.108) and (8.105) O  H 2 S → SO  H 2

(8.108)

SO  O2 → SO2  O

(8.105)

with some degree of chain branching by reaction (8.106) O  H 2 S → OH  SH

(8.106)

In the second stage, reaction (8.105) predominates over reaction (8.108) because of the depletion of the H2S; then the OH concentration rises via reaction (8.106) and begins the oxidation of the hydrogen O  H 2 → OH  H

(8.110)

Of course, the complete flame mechanism must include H  O2 → OH  O

(8.111)

OH  H 2 → H 2 O  H

(8.112)

Reactions (8.108) and (8.110) together with the fast reaction at the higher temperature SO  OH → SO2  H explain the known fact that H2S inhibits the oxidation of hydrogen.

(8.113)

448

Combustion

TABLE 8.4 Major Flame Chemistry Reactions of Sulfur under Rich Conditions (1)

H  S2  SH  S

(2)

S  H2  SH  H

(3)

SH  H2  H2S  H

(4)

S  H2S  SH  SH

(5)

OH  H2S  H2O  SH

(6)

H  SO2  SO  OH

(7)

S  OH  SO  H

(8)

SH  O  SO  H

(9)

H  SO2  M  HSO2  M

(10)

O  SO2  M  SO3  M

Using laser fluorescence measurements on fuel-rich H2/O2/N2 flames seeded with H2S, Muller et al. [43] determined the concentrations of SH, S2, SO, SO2, and OH in the post-flame gases. From their results and an evaluation of rate constants, they postulated that the flame chemistry of sulfur under rich conditions could be described by the eight fast bimolecular reactions and the two three-body recombination reactions given in Table 8.4. Reactions 1 and 6 in Table 8.4 were identified as the reactions that control the S2 and SO2 concentrations, respectively, while reactions 2 and 4 control the S concentration with some contribution from reaction 1. Reaction 6 was the major one for the SO flux rate. The three reactions involving H2S were said to play an important role. SH was found to be controlled by reactions 1–5. Because the first eight reactions were found to be fast, it was concluded that they rapidly establish and maintain equilibrium so that the species S, S2, H2S, SH, SO, and SO2 are efficiently interrelated. Thus relative concentrations, such as those of SO and SO2, can be calculated from thermodynamic considerations at the local gas temperature from the system of reactions SO2  H  SO  OH

(8.114)

OH  H 2  H  H 2 O

(8.112)

SO2  H 2  SO  H 2 O

(8.115)

The three-body recombination reactions listed in Table 8.4 are significant in sulfur-containing flames because one provides the homogeneous catalytic recombination of the important H2¶O2 chain carriers H and OH via (8.116) H  SO2  M  HSO2  M

449

Environmental Combustion Considerations

H  HSO2 → H 2  SO2

(8.117)

OH  HSO2 → H 2 O  SO2

(8.118)

SO2  O  M → SO3  M

(8.119)

while the other

is the only major source of SO3 in flames. Under rich conditions, the SO3 concentration is controlled by other reactions with H and SO, that is, SO  SO3 → SO2  SO2

(8.120)

H  SO3 → OH  SO2

(8.121)

and under lean conditions by O  SO3 → SO2  O2

(8.122)

b. COS and CS2 Even though there have been appreciably more studies of CS2, COS is known to exist as an intermediate in CS2 flames. Thus it appears logical to analyze the COS oxidation mechanism first. Both substances show explosion limit curves that indicate that branched-chain mechanisms exist. Most of the reaction studies used flash photolysis; hence very little information exists on what the chain-initiating mechanism for thermal conditions would be. COS flames exhibit two zones. In the first zone, carbon monoxide and sulfur dioxide form; and in the second zone, the carbon monoxide is converted into carbon dioxide. Since these flames are hydrogen-free, it is not surprising that the CO conversion in the second zone is rapidly accelerated by adding a very small amount of water to the system. Photolysis initiates the reaction by generating sulfur atoms COS  hv → CO  S

(8.123)

The S atom then engenders the chain branching step S  O2 → SO  O

21 kJ

(8.124)

which is followed by O  COS → CO  SO

213 kJ

(8.125)

SO  O2 → SO2  O

13 kJ

(8.105)

450

Combustion

At high temperatures, the slow reaction O  COS → CO2  S

226 kJ

(8.126)

must also be considered. Although the initiation step under purely thermally induced conditions such as those imposed by shocks has not been formulated, it is expected to be a reaction that produces O atoms. The high-temperature mechanism would then be reactions (8.105), and (8.124)–(8.126), with termination by the elimination of the O atoms. For the explosive reaction of CS2, Myerson et al. [44] suggested CS2  O2 → CS  SOO

(8.127)

as the chain-initiating step. Although the existence of the superoxide, SOO, is not universally accepted, it is difficult to conceive a more logical thermal initiating step, particularly when the reaction can be induced, in the 200–300°C range. The introduction of the CS by reaction (8.127) starts the following chain scheme: CS  O2 → CO  SO

347 kJ

(8.128)

SO  O2 → SO2  O

54 kJ

(8.105)

O  CS2 → CS  SO

326 kJ

(8.129)

CS  O → CO  S

322 kJ

(8.130)

S  O2 → SO  O

21 kJ

(8.131)

25 kJ

(8.132)

S  CS2 → S2  CS

O  CS2 → COS  S

(8.133)

O  COS → CO  SO

(8.126)

O  S2 → SO  S The high flammability of CS2 in comparison to COS is probably due to the greater availability of S atoms. At low temperatures, branching occurs in both systems via S  O2 → SO  O

(8.124)

Even greater branching occurs since in CS2 one has O  CS2 → CS  SO

(8.129)

Environmental Combustion Considerations

451

The comparable reaction for COS is reaction (8.125) O  COS → CO  SO

(8.125)

which is not chain branching.

c. Elemental Sulfur Elemental sulfur is found in the flames of all the sulfur-bearing compounds discussed in the previous subsections. Generally, this sulfur appears as atoms or the dimer S2. When pure sulfur is vaporized at low temperatures, the vapor molecules are polymeric and have the formula S8. Vapor-phase studies of pure sulfur oxidation around 100°C have shown that the oxidation reaction has the characteristics of a chain reaction. It is interesting to note that in the explosive studies the reaction must be stimulated by the introduction of O atoms (spark, ozone) in order for the explosion to proceed. Levy et al. [38] reported that Semenov suggested the following initiation and branching reactions: S8 → S7  S

(8.134)

S  O2 → SO  O

(8.124)

S8  O → SO  S  S6

(8.135)

with the products produced by SO  O → SO*2 → SO2  hv

(8.95)

SO  O2 → SO2  O

(8.105)

SO2  O2 → SO3  O

(8.136)

SO2  O  M → SO3  M

(8.137)

A unique feature of the oxidation of pure sulfur is that the percentage of SO3 formed is a very much larger (about 20%) fraction of the SOx than is generally found in the oxidation of sulfur compounds.

d. Organic Sulfur Compounds It is more than likely that when sulfur occurs in a crude oil or in coal (other than the pyrites), it is organically bound in one of the three forms listed in Table 8.3—the thiols, sulfides, or disulfides. The combustion of these compounds is very much different from that of other sulfur compounds in that

452

Combustion

a large portion of the fuel element is a pure hydrocarbon fragment. Thus in explosion or flame studies, the branched-chain reactions that determine the overall consumption rate or flame speed would follow those chains characteristic of hydrocarbon combustion rather than the CS, SO, and S radical chains which dominate in H2S, CS2, COS, and S8 combustion. A major product in the combustion of all organic sulfur compounds is sulfur dioxide. Sulfur dioxide has a well-known inhibiting effect on hydrocarbon and hydrogen oxidation and, indeed, is responsible for a self-inhibition in the oxidation of organic sulfur compounds. This inhibition most likely arises from its role in the removal of H atoms by the termolecular reaction H  SO2  M → HSO2  M

(8.116)

HSO2, a known radical that has been found in H2–O2–SO2 systems, is sufficiently inert to be destroyed without reforming any active chain carrier. In the lean oxidation of the thiols, even at temperatures around 300°C, all the sulfur is converted to SO2. At lower temperatures and under rich conditions, disulfides form and other products such as aldehydes and methanol are found. The presence of the disulfides suggests a chain-initiating step very similar to that of low-temperature hydrocarbon oxidation, RSH  O2 → RS  HO2

(8.138)

Cullis and Mulcahy reported that this step is followed by RS  O2 → R  SO2

(8.139)

to form the hydrocarbon radical and sulfur dioxide. One must question whether the SO2 in reaction (8.139) is sulfur dioxide or not. If the O2 strips the sulfur from the RS radical, it is more likely that the SO2 is the sulfur superoxide, which would decompose or react to form SO. The SO is then oxidized to sulfur dioxide as described previously. The organic radical is oxidized, as discussed in Chapter 3. The radicals formed in the subsequent oxidation, of course, attack the original fuel to give the RS radical, and the initiating step is no longer needed. The SH bond is sufficiently weaker than the CH bonds so that the RS radical would be the dominant species formed. At high temperatures, it is likely that the RS decay leads to the thioaldehyde H RS  M

R

HM

C S

(8.140)

453

Environmental Combustion Considerations

The disappearance of the thioaldehyde at these temperatures would closely resemble that of the aldehydes; namely, H R

X

C

RCS  XH

(8.141)

S i

M  R C S → R  CS  M

(8.142)

Then the CS radical is oxidized, as indicated in the previous discussion on CS2. The disulfide forms in the thiol oxidation from the recombination of the two RS radicals M  RS  RS → RSSR  M

(8.143)

The principal products in the oxidation of the sulfides are sulfur dioxide and aldehydes. The low-temperature initiating step is similar to reaction (8.138), except that the hydrogen abstraction is from the carbon atom next to the sulfur atom; that is, RCH 2 SCH 2 R  O2 → RCH 2 SßCHR  HO2

(8.144)

The radical formed in reaction (8.144) then decomposes to form an alkyl radical and a thioaldehyde molecule; that is, M  RCH 2 SßCHR → RCH 2  RCHS  M

(8.145)

Both products in reaction (8.145) are then oxidized, as discussed. The oxidation of the disulfides follows a similar route to the sulfide with an initiating step RCH 2 SSCH 2 R  O2 → RCH 2 SßSßCHR  HO2

(8.146)

followed by radical decomposition RCH 2 SßSCHßR → RCH 2 S  RCHS

(8.147)

The thiol is then formed by hydrogen abstraction RCH 2 S  RH → RCH 2 SH  R and the oxidation proceeds as described previously.

(8.148)

454

Combustion

e. Sulfur Trioxide and Sulfates As was pointed out earlier, the concentration of sulfur trioxide found in the combustion gases of flames, though small, is greater than would be expected from equilibrium calculations. Indeed, this same phenomenon exists in large combustors, such as furnaces, in which there is a sulfur component in the fuel used. The equilibrium represented by Eq. (8.94) SO2  21 O2  SO3

(8.94)

is shifted strongly to the left at high temperatures, so one would expect very little SO3 in a real combustion environment. It is readily apparent, then, that the combustion chemistry involved in oxidizing sulfur dioxide to the trioxide is such that equilibrium cannot be obtained. Truly, the most interesting finding is that the superequilibrium concentrations of SO3 are very sensitive to the original oxygen concentration. Under fuel-rich conditions approaching even stoichiometric conditions, practically no SO3 is found. In proceeding from stoichiometric to 1% excess air, a sharp increase in the conversion of SO2 to SO3 is found. Further addition of air causes only a slight increase; however, the effect of the excess nitrogen in reducing the temperature could be a moderating factor in the rate of increase. Figure 8.12, taken from the work of Barrett et al. [45] on hydrocarbon flames, characterizes the results generally found both in flame studies and in furnaces.

40

Range of values

SO3 (ppm)

30

20

SO3 Equivalent to 1% conversion of S to SO3

10 Sampling position: 24 in. from burner plate 0 95

105 115 125 Total air, percentage of stoichiometric air

FIGURE 8.12 Effect of excess air on the formation of SO3 in a hydrocarbon–air flame (after Barrett et al. [45]).

455

Environmental Combustion Considerations

Such results strongly indicate that the SO2 is converted into SO3 in a termolecular reaction with oxygen atoms: O  SO2  M → SO3  M

(8.119)

It is important to note that the superequilibrium results are obtained with sulfur fuels, small concentrations of sulfur fuels added to hydrocarbons, SO2 added to hydrocarbon, and so forth. Further confirmation supporting reaction (8.119) as the conversion route comes from the observation that in carbon monoxide flames the amount of SO3 produced is substantially higher than in all other cases. It is well known that, since O atoms cannot attack CO directly, the SO3 concentration is much higher in CO flames than in any other flames. The fact that in all cases the SO3 concentration also increases with pressure supports a termolecular route such as reaction (8.119). It is well known that the thermal dissociation of SO3 is slow and that the concentration of SO3 is therefore frozen within its stay time in flames and furnaces. The thermal dissociation rates are known, but one can also calculate the superequilibrium concentration of oxygen atoms in flames. If one does so, the SO3 concentration should correspond to the equilibrium concentration given by reaction (8.119), in which the oxygen atom superequilibrium concentration is used. However, the SO3 concentrations are never this high; thus, one must conclude that some SO3 is being reduced by routes other than thermal decomposition. The two most likely routes are by oxygen and hydrogen atom attack on the sulfur trioxide via O  SO3 → O2  SO2

152 kJ

(8.122)

H  SO3 → OH  SO2

79 kJ

(8.121)

Evidence supports this contention as well as the suggestion that reaction (8.122) would be more important than reaction (8.121) in controlling the SO3 concentration with reaction (8.119). Furthermore, one must recognize that reactions (8.119) and (8.122) are effective means of reducing the O radical concentration. Since reaction (8.116) has been shown to be an effective means of reducing H radical concentrations, one can draw the important general conclusion that SO2 and SO3 compounds reduce the extent of superequilibrium concentration of the characteristic chain-carrying radicals that exist in hydrocarbon flames. In furnaces using residual oils, heterogeneous catalysis is a possible route for the conversion of SO2 to SO3. Sulfur dioxide and molecular oxygen will react catalytically on steel surfaces and vanadium pentoxide (deposited from vanadium compounds in the fuel). Catalytic reactions may also occur at lower temperatures where the equilibrium represented by reaction (8.94) favors the formation of SO3.

456

Combustion

If indeed SO2 and SO3 are effective in reducing the superequilibrium concentration of radicals in flames, sulfur compounds must play a role in NO formation from atmospheric nitrogen in flame systems. Since SO2 and SO3 form no matter what type of sulfur compound is added to combustion systems, these species should reduce the oxygen atom concentration and hence should inhibit NO formation. Wendt and Ekmann [46] have reported flame data that appear to substantiate this conclusion. In examining reactions (8.119) and (8.122), one realizes that SO2 plays a role in catalyzing the recombination of oxygen atoms. Indeed, this homogeneous catalytic recombination of oxygen atoms causes the decrease in the superequilibrium concentration of the oxygen atoms. SO2 also plays a role in the recombination of hydrogen radicals through the route H  SO2  M → HSO2  M

(8.116)

H  HSO2 → H 2  SO2

(8.117)

and in the recombination of hydrogen and hydroxyl radicals through the route of reaction (8.116) and OH  HSO2 → H 2 O  SO2

(8.118)

Combining the preceding considerations of SO3 formation with the realization that SO2 and SO are the major sulfur oxide species in flames, one may use the term SOx to specify the sum of SO, SO2, and SO3. Also, considering the fact that in any combustion system sulfur can appear in the parent hydrocarbon fuel in various forms, it becomes evident that the details of the reaction mechanism for SOx formation are virtually impossible to specify. By virtue of the rapidity of the SOx formation process, Bowman [1] has argued that the need for a detailed fuel–sulfur oxidation model may be circumvented by postulating approximate models to estimate the gaseous SOx product distribution in the exhaust products. It has been suggested [46] that three principal assumptions would be involved in the proposed model: (1) the fuel–sulfur compounds should be considered minor species so that the major stable species concentrations would be those due to combustion of the hydrocarbon fuel; (2) the bimolecular H2–O2 reactions and reaction (8.115) would be partially equilibrated in the post-flame gases; (3) the SO3 concentration would be calculated from reactions (8.119), (8.121), and (8.122). The SOx pool would then vary as the overall reaction approaches equilibrium by means of the H2–O2 radical recombination reactions, the reactions that determine the equilibrium between SO2 and SO3, and those that affect the catalytic recombination of H, O, and OH.

f. SOx¶NOx Interactions The reactions of fuel–sulfur and fuel–nitrogen are closely coupled to the fuel oxidation; moreover, sulfur-containing radicals and nitrogen-containing radicals

Environmental Combustion Considerations

457

compete for the available H, O, OH radicals with the hydrocarbons [21]. Because of this close coupling of the sulfur and nitrogen chemistry and the H, O, OH radical pool in flames, interactions between fuel–sulfur and fuel– nitrogen chemistry are to be expected. The catalytic reduction of the radicals, particularly the O atom, by sulfur compounds will generally reduce the rates of reactions converting atmospheric nitrogen to NO by the thermal mechanism. However, experiments do not permit explicit conclusions [21]. For example, Wendt and Ekmann [46] showed that high concentrations of SO2 and H2S have an inhibiting effect on thermal NO in premixed methene–air flames, while deSoete [47] showed the opposite effect. To resolve this conflict, Wendt et al. [48] studied the influence of fuel– sulfur on fuel–NO in rich flames, whereupon they found both enhancement and inhibition. A further interaction comes into play when the thermal DeNOx process is used to reduce NOx. When stack gases cool and initial sulfur is present in the fuel, the SO3 that forms reacts with water to form a mist of sulfuric acid, which is detrimental to the physical plant. Furthermore, the ammonia from the thermal DeNOx process reacts with water to form NH4HSO2—a glue-like, highly corrosive compound. These SO3 conditions can be avoided by reducing the SO3 back to SO2. Under stack (post-combustion) temperatures, the principal elementary reactions for SO3 to SO2 conversion are HO2  SO3 → HSO3  O2 HSO3  M → SO2  OH  M Since the key to this sequence is the HO2 radical, the aqueous hydrogen peroxide process discussed for NO to NO2 conversion in the stack would be an appropriate approach. The SO2 forms no corrosive liquid mist in the stack and could be removed by wet scrubbing of the exhaust.

E. PARTICULATE FORMATION In earlier sections of this chapter, the role that particulates play in a given environmental scenario was identified. This section will be devoted exclusively to combustion-generated particulates whose main constituent is carbon. Those carbonaceous particulates that form from gas-phase processes are generally referred to as soot, and those that develop from pyrolysis of liquid hydrocarbon fuels are generally referred to as coke or cenospheres. Although various restrictions have been placed on carbon particulate emissions from different types of power plants, these particles can play a beneficial, as well as a detrimental, role in the overall plant process. The detrimental effects are well known. The presence of particulates in gas turbines can severely affect the lifetime of the blades; soot particulates in diesel engines absorb carcinogenic materials, thereby posing a health hazard. It

458

Combustion

has even been postulated that, after a nuclear blast, the subsequent fires would create enormous amounts of soot whose dispersal into the atmosphere would absorb enough of the sun’s radiation to create a “nuclear winter” on Earth. Nevertheless, particulates can be useful. In many industrial furnaces, for example, the presence of carbon particulates increases the radiative power of the flame, and thus can increase appreciably the heat transfer rates. The last point is worth considering in more detail. Most hydrocarbon diffusion flames are luminous, and this luminosity is due to carbon particulates that radiate strongly at the high combustion gas temperatures. As discussed in Chapter 6, most flames appear yellow when there is particulate formation. The solid-phase particulate cloud has a very high emissivity compared to a pure gaseous system; thus, soot-laden flames appreciably increase the radiant heat transfer. In fact, some systems can approach black-body conditions. Thus, when the rate of heat transfer from the combustion gases to some surface, such as a melt, is important—as is the case in certain industrial furnaces—it is beneficial to operate the system in a particular diffusion flame mode to ensure formation of carbon particles. Such particles can later be burned off with additional air to meet emission standards. But some flames are not as luminous as others. Under certain conditions the very small particles that form are oxidized in the flame front and do not create a particulate cloud. It is well known that the extent of soot formation is related to the type of flame existing in a given process. Diesel exhausts are known to be smokier than those of spark-ignition engines. Diffusion flame conditions prevail in fuel-injected diesel engines, but carbureted spark-ignition engines entail the combustion of nearly homogeneous premixed fuel–air systems. The various phenomena involved in carbon particulate formation have been extensively studied. The literature is abundant and some extensive review articles [49–51] are available. Most of the subsequent material in this chapter will deal with soot formation while a brief commentary on the coke-like formation from liquid fuels will be given at the end.

1. Characteristics of Soot The characteristics of soot are well described in the article by Palmer and Cullis [49], who provide detailed references on the topic. Aspects of their review are worth summarizing directly. They report the detailed physical characteristics of soot as follows: The carbon formed in flames generally contains at least 1% by weight of hydrogen. On an atomic basis this represents quite a considerable proportion of this element and corresponds approximately to an empirical formula of C8H. When examined under the electron microscope, the deposited carbon appears to consist of a number of roughly spherical particles, strung together rather like pearls on a necklace. The diameters of these particles vary from 100 to 2000 Å and most commonly lie between 100 and 500 Å. The smallest particles are found in luminous but nonsooting flames, while the largest are obtained in heavily sooting flames. X-ray diffraction shows that each particle is made up of a large

Environmental Combustion Considerations

459

number (104) of crystallites. Each crystallite is shown by electron diffraction to consist of 5–10 sheets of carbon atoms (of the basic type existing in ideal graphite), each containing about 100 carbon atoms and thus having length and breadth of the order of 20–30 Å. But the layer planes, although parallel to one another and at the same distance apart, have a turbostratic structure, that is, they are randomly stacked in relation to one another, with the result that the interlayer spacing (3.44 Å) is considerably greater than in ideal graphite (3.35 Å). It may readily be calculated on this picture of dispersed carbon deposits that an “average” spherical particle contains from 105 to 106 carbon atoms.

Investigators have used the words “carbon” and “soot” to describe a wide variety of carbonaceous solid materials, many of which contain appreciable amounts of hydrogen as well as other elements and compounds that may have been present in the original hydrocarbon fuel. The properties of the solids change markedly with the conditions of formation; and, indeed, several quite well-defined varieties of solid carbon may be distinguished. One of the most obvious and important differences depends on how the carbon is formed: carbon may be formed by a homogeneous vapor-phase reaction; it may be deposited on a solid surface that is present in or near the reaction zone; or it may be generated by a liquid-phase pyrolysis.

2. Soot Formation Processes Determining the relative tendency of hydrocarbon fuels to soot, explaining why this relative tendency between fuels exists, and discovering how to control or limit soot production in a particular combustion process—these are the elements of greatest importance to the practicing engineer. Since the amount of soot formed from a particular fuel has a complex dependence on the overall combustion process, no single characteristic parameter can define the amount formed per unit weight of fuel consumed. Both the flame type and various physical parameters determine the extent of soot formation from a given fuel. Moreover, depending upon the combustion process, the final mass of particulates emitted from the system could be reduced by a particle afterburning process. Examining the detailed chemical processes of soot formation and oxidation, one notes how very complex the overall system is. Regardless of the flame type, the fuel undergoes either pure or oxidative pyrolysis. If the fuel is nonaromatic, some precursors of the fuel pyrolysis undergo cyclization to create an aromatic ring. The ring structure is thought to add alkyl groups, developing into a polynuclear aromatic (PAH) structure that grows owing to the presence of acetylene and other vapor-phase soot precursors. Under certain flame conditions, the elements forming large aromatic structures are simultaneously oxidized. The precursors and all subsequent structures must be conjugated so that they are resonancestabilized at the high temperatures in which they grow. Eventually, the aromatic structures reach a large enough size to develop into particle nuclei. Such condensed-phase carbon particles contain large amounts of hydrogen. The particles dehydrogenate (age) in a high-temperature combustion field while

460

Combustion

physically and chemically absorbing gaseous hydrocarbon species that dehydrogenate, thereby effecting a large increase of soot mass. The growing particles also agglomerate and conglomerate. The absorbed species, to a large degree, undergo chemical reformation, which results in a carbonaceous soot structure. And while these events are occurring, oxidative attack on the particles continues to form gaseous products. Nevertheless, the eventual chemical steps in the soot formation process appear to be similar, regardless of the initial fuel. As Palmer and Cullis [49] have noted, “With diffusion flames and premixed flames investigations have been made of the properties of the carbon formed and of the extent of carbon formation under various conditions. In general, however, the properties of the carbon formed in flames are remarkably little affected by the type of flame, the nature of the fuel being burnt, and the other conditions under which they are produced.” Although many current investigations have shown that the physical and chemical processes noted above are both complex and various, the generalized model to be described below allows one to identify certain controlling steps, thus gaining some important insights into the relative tendency of fuels to soot under particular flame configurations. And, using this insight, one may develop means to control the amount of particulate emission.

3. Experimental Systems and Soot Formation Estimates of fuel sooting tendency have been made using various types of flames and chemical systems. In the context to be used here, the term sooting tendency generally refers to a qualitative or quantitative measure of the incipient soot particle formation and growth rates of one fuel relative to another in a particular experimental combustion configuration. In many cases, this tendency varies strongly with the type of flame or combustion process under investigation. This variation is important because the incipient soot particle formation and growth rates determine the soot volume fraction formed and emitted in a combustion system. For premixed fuel–air systems, results are reported in various terms that can be related to a critical equivalence ratio at which the onset of some yellow flame luminosity is observed. Premixed combustion studies have been performed primarily with Bunsen-type flames [52, 53], flat flames [54], and stirred reactors [55, 56]. The earliest work [57, 58] on diffusion flames dealt mainly with axisymmetric coflow (coannular) systems in which the smoke height or the volumetric or mass flow rate of the fuel at this height was used as the correlating parameter. The smoke height is considered to be a measure of the fuel’s particulate formation and growth rates but is controlled by the soot particle burnup. The specific references to this early work and that mentioned in subsequent paragraphs can be found in Ref. [50]. Work on coflowing Wolfhard–Parker burners [59, 60], axisymmetric inverse coflowing configurations (oxidizer is the central jet) [61, 62], and counterflow

Environmental Combustion Considerations

461

(opposed-jet) diffusion flames [63–65] has employed chemical sampling and laser diagnostics for scattering, extinction, and fluorescence measurements to determine chemical precursors, soot number density, volume fraction, and average particle size. These experiments not only obtained a measurement of where and when incipient soot formation takes place in a particular flame configuration, but also followed the variation of the number density, volume fraction, and size with time after incipient formation. An important feature of most of these flame configurations is that they separate any particle burnup from the formation rate. As described in Chapter 4, counterflow experiments are designed to be directly opposing gas jets; a flow stream, normally the oxidizer, opposing the fuel flow emanating from a horizontal porous cylinder; or a directly opposing jet emanating from a cylindrical tube. Thus the porous cylinder condition creates flames that emulate the wakes of burning liquid droplets in various types of streams. A small liquid droplet burning in perfect spherical symmetry is nonluminous; larger droplets, because of a longer diffusion time from the liquid surface to the flame front, facilitate fuel pyrolysis and growth and have mildly luminous flames. In another approach, shock tube experiments in which fuel pyrolysis leads to soot also provide some means of measuring the relative sooting tendency, but not necessarily applicable to a combustion process. Confusion can arise, however, in interpreting soot formation results unless one understands how the structures of the processes taking place in premixed flames, normal coflowing, inverse coflowing, and counterflowing diffusion flames; or shock tubes differ. This point cannot be overemphasized. In premixed flames the formation of soot precursors through fuel pyrolysis is competitive with oxidative attack on these precursors, but no oxidative attack occurs on the precursors formed during the fuel pyrolysis in diffusion flames. However, in normal coflowing diffusion flames, the fuel stream, its pyrolysis products, and other constituents added to it are convected by fluid motion across the flame front; that is, convection dominates molecular diffusion. These constituents flow into an environment of increasing temperature. Recall that, for a pure fuel jet (a mass fraction of 1) in counterflow diffusion flames, all elements in the fuel stream reach the flame front only by molecular diffusion across the stagnation plane. Soot particles form only on the oxidizer side of the flame front in a pure air–fuel system; then they are convected toward the stagnation plane and are not oxidized. Unfortunately, this counterflow procedure does not occur in any industrial combustion process. The flow pattern in inverse axisymmetric coflowing flames is such that the precursors and particulates formed are also convected away from the flame zone. In shock tubes, fuel pyrolysis (and soot formation) takes place at a relatively fixed temperature and pressure. Thus, it is possible to measure a relative sooting parameter at a fixed temperature. Although stoichiometric temperatures similar to the temperatures reached in shock tubes can be established with coflowing diffusion flames, soot formation in these flames occurs at much lower temperatures prior to reaching the stoichiometric flame surface. In diffusion flames the

462

Combustion

temperature–time history of a fuel element leading to pyrolysis, particulate formation, and growth varies; in shock tube experiments it generally does not. Flame turbulence should not affect soot formation processes under premixed combustion conditions, and the near correspondence of the results from Bunsen flames [52] and stirred reactors [55] tends to support this contention.

4. Sooting Tendencies Most of the early work and more recent efforts on soot formation under premixed conditions were conducted with air as the oxidizer. These early results reported data as a critical sooting equivalence ratio φc, in which the higher the φc, the lower the tendency to soot. As shown in Table 8.5, the trend observed followed the order Aromatics Alkanes Alkenes Alkynes that is, ethane was reported to have a greater sooting tendency than ethene or acetylene. The most extensive early data of sooting under laminar diffusion flame conditions, as measured by the smoke height method, were obtained by Schalla et al. [57, 58]. The general trend observed followed the order Aromatics Alkynes Alkenes Alkanes Upon comparison, the trends of the alkyl compounds in the premixed case revealed an inconsistency with the oxidation reaction mechanism of ethene under fuel-rich conditions: it was found that the alkane was rapidly converted first to the alkene and then to the alkyne [66, 67]. Thus one would expect the alkynes to exhibit the greatest sooting propensity under premixed conditions. The answer to this inconsistency was developed from the early work of Milliken [68], who studied the soot emission from ethene on a cooled flat flame burner. Milliken found that the cooler the flame, the greater the tendency to soot (i.e., the lower the critical sooting equivalence ratio). He explained this temperature trend by evaluating the effect of temperature on the two competing processes occurring in the sooting flame: the pyrolysis rate of the fuel intermediate (acetylene precursor) leading to the other precursors and the rate of oxidative (hydroxyl radical, OH) attack on these precursors. Both rates increase with temperature, but the oxidative rate increases faster. Thus the tendency of premixed flames to soot diminishes as the temperature rises. Other investigators [67, 69, 70] verified Milliken’s calculations and extended the concept by showing that in coflow diffusion flames, where there is no oxidative attack on the precursors, the soot volume fraction created increases with temperature. Therefore, for a proper comparison of the effect of fuel structure on the sooting tendency, the flame temperature must be controlled. As will be noted

463

Environmental Combustion Considerations

TABLE 8.5 Critical Sooting Equivalence Ratios (φc) of Various Fuels Premixed with Air φc Fuel

[52]

Ethane

1.67

[54]

Propane

1.56

n-Pentane

1.47

Isopentane

1.49

n-Hexane

1.45

Isohexane

1.45

1.73

n-Octane

1.39

1.73

Iso-octane

1.45

n-Cetane

1.35

Acetylene

2.08

Ethylene

1.82

Propylene

1.67

Butylene

1.56

Amylene

1.54

Cyclohexane

1.52

[56]

[53] 1.70

1.73

1.61

1.70

2.00

2.00 2.00

1.74

1.70

n-Heptene

1.45

Ethyl alcohol

1.52

Propyl alcohol

1.37

Isopropyl alcohol

1.39

Isoamyl alcohol

1.54

Acetaldehyde

1.82

Propionaldehyde

1.75

Methyl ethylketone

1.69

Diethyl ether

1.72

Diisopropyl ether

1.56

Benzene

1.43

1.54

Toluene

1.33

1.46

1.39

Xylene

1.27

1.45

1.30

Cumeme

1.28

1.56

1.40

Methyl naphthalene

1.03

1.21

1.75

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Combustion

1.5 With “air” 1.4

1.3

Ethylene

Ethane

Propene 1.2

Propane

1.1 ψc

1-Butene Acetylene 1,3-Butadiene 1-Hexene

Butane

2-Methylpropane 1.0 2-Methylpropene Hexane Cyclohexane Octane

.9

.8

Cyclohexene 1-Octene

Benzene Toluene Cumene 1,2,3,4,-Tetrahydronaphthalene

.7 1- Methylnaphthalene .6 1.7

1.8

1.9

2.0 Tf 

2.1

2.2

2.3

2.4

2.5

103 (K)

FIGURE 8.13 Critical sooting equivalence ratios (ψc) based on CO and H2O stoichiometry of various hydrocarbon fuels as a function of calculated adiabatic flame temperature Tf.

subsequently, it is the temperature field created by the diffusion flame that is more important than the actual flame temperature [70a]. Early work on premixed Bunsen flames and coflowing diffusion flames [69–71] was repeated in experiments where the temperature was controlled by varying the N2/O2 ratio in the oxidizer used for the Bunsen experiments and by adding inerts, particularly nitrogen, to the fuel in the diffusion flame experiments. The critical sooting equivalence ratio in premixed flames was determined by the visual observation of the onset of soot luminosity. Varying the nitrogen concentration permitted the determination of a critical sooting equivalence ratio ψc (whereas φc is based on a stoichiometry referenced to carbon dioxide and water as products; ψc is based on a stoichiometry referenced to carbon monoxide and water as products since all conditions of operation are very fuel rich) as a function of the calculated adiabatic flame temperature. The results are shown in Fig. 8.13. The values corresponding to air agree exactly with the early work of Street and Thomas [52]. However, one will note that at

465

Environmental Combustion Considerations

1.5 Ethane 1.4 Propane Ethylene 2-Methylpropane Butane Propene 1.2 2-Methylpropene 1-Butene 1.1 Hexane Acetylene Cyclohexane  1,3-Butadiene Octane 1.0 1-Hexene

ψc

1.3

0.9 0.8

Cyclohexene 1-Octene Benzene Toluene Cumene

0.7

1,2,3,4-Tetrahydronaphthalene 0.6 1-Methylnaphthalene 0.5 0

2

4

6

8

10

12

14

16

18

Number of C – C bonds FIGURE 8.14 Critical sooting equivalence ratio ψc at 2200 K as a function of the “number C¶C bonds” in hydrocarbon fuels. , 0, and  indicate ethane/l-octane mixtures in molar ratios of 5 to 1, 2 to 1 and 1 to 2, respectively; χ, acetylene/benzene at a molar ratio of 1 to 3. The O symbol for 2 to 1, falls on top of the butene symbol.

a fixed temperature, ψc (acetylene) ψc (ethene) ψc (ethane), as one would expect. Also, there is no specific trend with respect to homologous series. Following the conceptual idea introduced by Milliken [68], Takahashi and Glassman [53] have shown, with appropriate assumptions, that, at a fixed temperature, ψc could correlate with the “number of C¶C bonds” in the fuel and that a plot of the log ψc versus number of C¶C bonds should give a straight line. This parameter, number of C¶C bonds, serves as a measure of both the size of the fuel molecule and the C/H ratio. In pyrolysis, since the activation energies of hydrocarbon fuels vary only slightly, molecular size increases the radical pool size. This increase can be regarded as an increase in the Arrhenius pre-exponential factor for the overall rate coefficient and hence in the pyrolysis and precursor formation rates so that the C/H ratio determines the OH concentration [72]. The ψc versus C¶C bond plot is shown in Fig. 8.14. When these

466

Combustion

data are plotted as log ψc versus C¶C bonds, a straight line is obtained [53]. Also plotted on Fig. 8.14, as a test of the correlation, are the results of some fuel mixtures [73]. These data lead to the important conclusion that specific fuel structures do not play a role under premixed conditions; for example, decane has the same sooting tendency as benzene at the same premixed flame temperature. Thus, under premixed conditions, all fuels must break down to the same essential species that build into a soot particle. This postulate is consistent with the results of Harris and Weiner [74, 75], who showed that increments of carbon introduced as ethene in premixed flames were just as effective in producing soot as increments of carbon introduced as toluene and that the greatest mass of soot came about by growth on particles in which the dominant species contributing to this particle growth was acetylene. Similar temperature-control procedures were applied to a highly overventilated concentric coflowing diffusion flame [51, 76] in which 29 fuels were evaluated with respect to their sooting tendency. In these experiments the mass flow rate of the fuel jet is increased until soot breaks through the top of the “luminous visible flame front.” This “luminous visible flame front” is not the flame; its edge defines the region in which the soot particles formed were completely burned. As stated earlier the stoichiometric (Burke–Schumann) flame front that defines the flame temperature lies within this luminous envelope. The initial breakthrough of this luminous boundary is referred to as the sooting or smoke height, and most data are reported in terms of the fuel mass flow rate at this height. Indeed the initial breakthrough does not occur at the fuel jet axis, but in a toroidal ring around the axis. A planar view of a smoke height condition appears as two “wings.” This ring develops because the flow streamlines in the axisymmetric coflow approach are not parallel. If nitrogen, or any inert species, is added to the fuel jet when the smoke height is reached, the luminous zone closes and soot no longer emanates from the top of the flame. If fuel mass flow is then increased, another smoke height is reached. Additional inert diluent again suppresses the soot emanation and additional fuel flow is necessary for the flame to smoke. In this manner the smoke height, or mass flow rate at the smoke height, of a single fuel may be obtained at different flame temperatures since the diluent lowers the temperature. Whether the effect of diluent on soot exhaust through a flame front is due mainly to temperature variation or not is also discussed subsequently How this smoke effect varies with inert addition is best explained by considering the results of many early investigators who reported that incipient soot formation occurred in a very narrow temperature range. The various results are shown in Table 8.6. Since, as stated earlier, the incipient particle formation mechanisms for various fuels follow quite similar routes, it seems appropriate to conclude that a high activation energy process or processes control the incipient particle formation. The best concept and evidence to date is that given by Dobbins [77]. It is likely that the slight variation of temperatures shown in Table 8.6 is attributable to the different experimental procedures

467

Environmental Combustion Considerations

TABLE 8.6 Soot Inception Temperature (K) by Various Investigators Fuel

A

Methane

1913

Propane

1783

Ethene

1753

n-Butane

1723

Cyclobutane

1643

Butene

1623

Acetylene

B

C

D

E

1750

1460

1684

1348

1600

1665

1390

1650

1353

Butadiene

1623

1650

Allene

1613

1586

Benzene

1580

Toluene

1570

1750 1332

F

1430 1750

A. Dilution of fuel jet until no luminosity [Smith, C. A., M.S.E. Thesis, Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey (1990)]. B. Dilution of fuel jet until no luminosity (Glassman et al. [77]). C. Centerline observation of first particulates (Gomez et al. [86]). D. Optical measurements of inception and temperature [Boedecker, L. R., and Dobbs, G. M., Proc. Combust. Inst. 21, 1097 (1986)]. E. Critical sooting C/O ratio in premixed flames as a function of controlled flame temperature [Bohm, H., Hesse, D., Jander, H., Luers, B., Pietscher, J., Wagner, H. Gg., and Weiss, M., Proc. Combust. Inst. 22, 403 (1988)]. F. Flat premixed flame observations and measurements [Olson, D. B., and Madronvich, S., Combust. Flame 60, 203 (1985)].

used. The experimental procedure of Ref. [78] sheds some light on the concept proposed here. In these experiments, a given sooting flame was diluted until all particle radiation just disappeared, then the temperature of the flame front was measured on the centerline of the coannuluar configuration. These measurements are shown in Figs. 8.15 and 8.16. Since different amounts of dilution were required to eliminate particle continuum radiation for the different fuels, it should not be surprising that there is some variation for different fuels in the temperature limit for incipient particle formation. If the temperature limit is taken as the average of those reported in Figs. 8.15 and 8.16 (1650 K), it can be assumed that incipient particle formation occurs along a 1650 K isotherm in the coannular diffusion flame structure. Then, at the initial smoke height with no inert addition, the 1650 K isotherm within the fuel flow designates the point of incipient particle formation. Thus there is enough growth along flow streamlines that coincides with and closely parallels the 1650 K isotherm to cause particles to penetrate the flame and smoke. This flow coincidence will, of course, be near the sides of

468

Combustion

C2H2

100 20

Temperature (K) C4H6

15 100 20 10 15

Fuel rate Acetylene 3 H 1.45 cm 1.29 cm /s N2 gas rate 20.8 cm3/s

5

1,3-Butadiene H 1.54 cm Fuel rate 0.56 cm3/s N2 gas rate 22.6 cm3/s

10 5 0

0

0.5

1.0 1.5 Axial location (z/H )

2.0

FIGURE 8.15 Typical temperature profiles along the centerline of laminar hydrocarbon fuel jets diluted with N2 to the point of no luminosity when burning in overventilated air streams. H is the height of the flame; Z is the distance from jet exit along the centerline. 2000

1800 1-Butene

1800

1600

Ethene

1600 5.00

10.00

15.00

1800

20.00

1400

0

2.00

4.00

6.00

8.00

1800 Allene 1600

1600

1,3-Butadiene 1400

1400

0

2.00

4.00

6.00

NDF All blue flame B0

8.00 1200 1000

0

2.00

Diluted gas Ar N2 CO2 4.00

6.00

8.00

FIGURE 8.16 Incipient soot formation temperature as a function of the amount of diluent. The ordinate is the temperature in kelvins; the abscissa is f  (Qf/Qf  Qdg)  100, where Qf is the volumetric flow rate of the fuel and Qdg is that of the diluent.

the flame. It is for this reason that the smoke height is observed as a flame breakthrough, lending a wing-like appearance to the flame luminous envelope [51]. Thus what determines the smoke height is the growth in mass of the incipient particles created at about the 1650 K isotherm. When the fuel is diluted with an inert, the flame temperature drops; and since the situation represents a moving-boundary (quasi-steady) heat transfer problem, the 1650 K isotherm moves closer to the flame temperature isotherm. As a result, particle growth diminishes and the flame front closes. That is, because the mass of soot formed is smaller, it is consumed in the flame front, so no smoke height is observed. However, if the fuel mass is increased, more incipient particles form and, although growth is slower under dilute conditions, the soot mass again

Environmental Combustion Considerations

469

becomes large enough to exhibit a smoke height. By repeating the dilution procedure to further decrease the flame temperature, as stated, one can eliminate the smoke penetration, which can be induced again by adding more fuel. Consider, for example, the fact that a coannular diffusion flame with acetylene will have a much higher flame temperature than that of butene. When considered at the centerline, the incipient particle temperature isotherm for acetylene is much further from the flame temperature isotherm at the same position than is that of butene; that is, the acetylene particles formed have a larger growth path. Thus, although more incipient particles may form at about 1650 K for butene, the overall soot volume fraction from acetylene may be greater due to a longer growth period. The depth of the thermal wave (or isotherm) is of the order α/u where α is the thermal diffusivity and u is the linear velocity. There are self-compensating terms in each of these parameters even when there is dilution. Recall that the flame height for a diluted fuel varies to the first order solely with the volumetric flow rate of the fuel, even under buoyancy conditions. Figure 8.16 shows some results of this soot elimination procedure in which diluents other than nitrogen were chosen. Argon requires greater dilution to achieve the same temperature as nitrogen, and carbon dioxide requires less. Yet the variation in the incipient particle temperature for the four fuels shown in Fig. 8.16 is slight and well within experimental error. Because different fuels produce different amounts of incipient soot particles at about 1650 K, the smoke height test becomes a measure of the fuel’s relative propensity to soot under flame-like conditions. However, if the smoke height is to be used for fuel comparisons, the diffusion flame temperature must be adjusted to be the same for all fuels. Further, based on the results of Harris and Weiner [74, 75] for premixed flames, the assumption is made that the growth rate of particles is similar for all fuels that produce particles. Opposed-jet systems also permit good estimates of particle growth times along the jet stagnation streamline as shown by the results of Vandsburger et al. [63, 64], which reveal, as expected, that increasing the temperature (increasing the oxygen mole fraction in the opposed stream, i.e., the oxygen index) increases the soot yield and that ethene produces more soot than propane. The same variational trend in soot volume fraction with time was obtained with Wolfhard–Parker burners [60] as in premixed flames [74, 75]. Particle number density trends are also the same for various burner systems. For similar fuels, the general agreement between coflow and counterflow results for soot volume fractions, number densities, and particle size ranges is reasonably good. Since residence time is reduced near the flame in counterflow burners in comparison with coflow systems, the maximum soot formation rates in counterflow systems are found to be somewhat lower than those in coflow systems for comparable fuels and flame temperatures. When soot growth rates are normalized by the available soot surface area to give what Harris and Weiner [74, 75] call the “specific surface growth rate,” the results of coflow burners agree not only with those of counterflow burners, but also with those of rich premixed flames [51].

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Combustion

Detailed optical measurements of the growth patterns both in coflow and counterflow diffusion flame configurations have provided great insight into the extent of particle growth after soot nucleation. Many research efforts in these configurations have been performed in order to evaluate how additives to the fuel jet would affect soot formation and growth by causing either flame temperature variations or chemical inhibition. Considering that most of these efforts used optical diagnostics, it is important to note that due to the work of Axelbaum and Law [79] it was shown that these type of additive measurements had a greater effect on soot volume fraction developed than flame temperature control; that is, the additives affected the measurement rather than any soot process. This concept was verified provided the resulting temperature was above the nucleation temperature. Of course, as discussed earlier, for diffusion flames, when the flame temperature is below the nucleation temperature (1650 K at 1 atm), the kinetic time scales for soot nucleation are too slow for particulate formation. In the context stated earlier, what controls the soot volume fraction that penetrates the laminar diffusion flame to cause the smoke height is the distance between the isotherms that specify the incipient particle formation (nucleation) and the stoichiometric flame temperature. Again, this distance establishes the growth time of the particles formed before flame oxidation of the soot occurs, or in the case of additives allows soot volume fraction measurements to evaluate the possible effect of these additives on soot mass fraction trends. As stated, what exists for the given gaseous flow rate in diffusion flames is a quasisteady heat transfer condition which determines the temperature field within the burning, soot forming, jet. Consequently the thermal diffusivity of the flowing gas mixture and its velocity are of direct importance to the overall soot volume fraction formed within the flame structure. Now considering conditions along the diffusion flame centerline streamline, then a simple one-dimensional moving-boundary problem exists and the temperature variation along the centerline (streamline) can be readily shown [80] to be ln [(Tf  To )/(Tn  To )]  (u/α )z

(8.148a)

where Tf, Tn and To are the flame, incipient particle formation, and the initial fuel jet temperatures, respectfully, u is the velocity along the axial streamline and z the distance between Tf and Tn or the particle growth distance within the diffusion flame, (α/u) is generally referred to as the depth of the thermal wave (field). The centerline is, of course, the most appropriate and simplest streamline to analyze. In a conical diffusion flame all other streamlines curve. Streamlines emanating near the wall of the fuel jet will curve to the extent that Tf and Tn essentially merge so there is an area of blue luminosity (no soot). What is important to realize from Eq. (8.148a) is that the flame temperature effect is in the log term and thus variations in flame temperature are not significant with regard to increases of soot mass in diffusion flames. The extent of the

Environmental Combustion Considerations

471

thermal field is important; that is, the depth of the thermal wave (α/u) inside the burning fuel jet. Although in turbulent jet flames the flame isotherm is fluctuating and thus is not spatially fixed, but within a band, lifted or not, the general trend given by Eq. 8.148a has been found to be appropriate by experiments conducted [81]. In essence, then, physical conditions which affect fuel stream velocity and thermal diffusivity and their dependency on pressure and jet exit temperature are what dominate the extent of soot production in diffusion flame situations. Most early diffusion flame experimental work with respect to soot tendency was under small Froude number consideration, but has given insight into the laminar flame sooting process. Also, one must keep in mind that under the buoyancy condition the jet fuel exit velocity increases as the actual flame height is approached. Sooting flame experiments under microgravity conditions, which in essence is a completely momentum-controlled situation based on fuel exit velocity, show a bellowing flame structure as that described in Fig. 6.9, which is a result of the Burke–Schumann development. It follows then that jet flames will soot more and appear wider in microgravity as has been found [81a]. As well, it should be noted that opposed jet diffusion flame work gives soot particle growth in an inverted temperature field compared to that given by Eq. 8.148a. Consequently, the growth patterns in opposed jet diffusion flames, and for that matter in inverted coflow annular flames, will not emulate fuel jet injection soot characteristics. The data of Axelbaum and Law [79] clearly supports the concept of the thermal field effect and the effect of inert dilution on soot volume fraction measurements as developed by Eq. 8.148a. Shown in Fig. 8.17 are their soot volume fraction data for a 100% ethene jet (A), 50% ethene–50% nitrogen jet (B) and a 50–50 jet (C) whose air was adjusted to give the same temperature as (A). Notice the close correspondence between (B) and (C) even though the flame temperatures are vastly different. The effect of thermal diffusivity is best shown in some early soot measurements [82] as given by Fig. 8.18. This plot correlates the ethene flow rate to reach the smoke height as a function of the added diluent flow rate. A lower ethene flow rate indicates a greater tendency to soot. At a fixed diluent flow rate that indicates a fixed average jet velocity, the results of Ar and N2 are practically the same even though the flame temperature would be appreciatively higher for Ar. Interestingly the thermal diffusivity of argon and nitrogen are practically the same. Now note the difference in the Ar and He results where the temperatures are the same, but the thermal diffusivity of He is an order of magnitude greater than Ar. The same aspect explains the difference between SO2 and CO2 in which the CO2 has a thermal diffusivity about a factor of 2 larger than SO2. Indeed the concept exemplified by Eq. (8.148a) explains much of the additive results in the literature. Remarkably, the simple smoke height experiments reveal the same trends without the necessity of the more complex soot volume fraction measurements. Of course, experiments to control the diffusion flame temperature by addition of inerts such as

472

Combustion

Integrated soot volume fraction  106 (cm)

0

4

8 A

C 2

6

B

0 0

10

20

30 40 Axial position (mm)

50

60

70

FIGURE 8.17 Effect of dilution and temperature reduction, resulting from inert addition to the fuel, on soot formation. A: 100% Ethene, B: 50% Ethene/50% Nitrogen, C: same as B except temperature has been adjusted to be equal to that of A by replacing a portion of the nitrogen in the oxidizer with an equal molar amount of argon. From Ref. [79].

helium to the air stream will produce results different in experiments when the inert is added to the fuel stream. The reasoning is that addition to the fuel stream affects the overall thermal diffusivity in the thermal field within the fuel stream. For example, if the nitrogen from the air stream is transferred to the fuel stream, the same stoichiometric temperature would prevail in a fuel jet flame. However, the additional volumetric flow of nitrogen will increase the fuel stream flow and decrease the depth of the thermal field (α/u). Thus, the extent of soot growth is reduced. The high oxygen content of the outer flow does not affect the flame structure chemistry, as some have claimed, and thus the soot mass development. Exhaust gas recirculation in various furnaces and diesel operations has been used in order to control NOx and soot formation. With regard to soot formation the question arises as to the effect of induced oxygen or oxygen-containing species on the soot forming process. It is known that even in simple laminar fuel jets, there are some very slight amounts of oxygen infused into the fuel jet stream. It has been shown in fuel jet experiments that small amounts of oxygen only have an effect on tightly bound simple fuels such as ethene or ethyne, otherwise oxygen acts as a diluent until very appreciable amounts enter the fuel stream [65]. The data shown in Fig. 8.19 use peak extinction coefficients as a soot measurement on cylindrical opposed jet diffusion flames. The oxygen index is that which set the adiabatic flame temperature for all equivalence

473

Environmental Combustion Considerations

8

Ethylene flow rate [cm3 s1]

7

6

5 SO2 CO2 H2O N2

4

Ar He

3

0

1

2

3 4 Diluent flow rate [cm3 s1]

5

6

7

FIGURE 8.18 The effect of inert diluents on sooting volume fraction. From Ref. 82

ratios on the fuel side of the opposed jet diffusion flame. Notice for ethene that both for an oxygen index of 0.21 (air equivalent) and richer (0.18) that the addition of oxygen on the fuel side increases product formation as it accelerates ethene pyrolysis and oxidation and heat release. Then the mixture becomes essentially premixed on the fuel side and as the fuel side temperature increases, the soot production decreases. For propane, however, addition of oxygen on the fuel side acts as a diluent until there is enough oxygen to induce chemical energy release. The oxygen index for the propane results was taken as 0.25 because propane does not have a smoke height in air. Smoke height results for ethene, propane, and isobutene correlate well with opposed jet results [65]. Most practical fuels have many constituents, which would indicate that some components would follow the trend indicated in Fig. 8.19. Indeed propane could be a simple surrogate fuel for more complex hydrocarbons in these type experiments. Thus the analysis indicated in Eq. (8.148a) would to a large extent correctly correlate results in high pressure diesel like chambers as given in Ref. [83]. In direct fuel injection diesel engine experiments the flame front lifts from the jet outlet and one must realize that this condition does not affect the soot analysis as given by Eq. (8.148a) since Tn and Tf lie well within the lifted flame

474

Combustion

0.5 Symbol

Flame Ethene. Ethene. Propane.

fw  1.45, fw  1.45, fw  2.0,

Oxygen index  0.21 Oxygen index  0.18 Oxygen index  0.25

Peak extinction coefficient (1/cm)

0.4

0.3

0.2

0.1

0

1

10



Equivalence ratio FIGURE 8.19 Peak extinction coefficient versus equivalence ratio of fuel/oxygen stream mixture in propane and ethene opposed-jet diffusion flames. fw is the fuel injection parameter. The greater the extinction coefficient, the greater the soot mass. From Ref. [93].

structure. Indeed, application of the principle of Eq. (8.148a) to turbulent lifted diesel flame-type experiments would explain many results reported in Ref. [83].

5. Detailed Structure of Sooting Flames The critical sooting equivalence ratios for premixed flames and the smoke heights for diffusion flames discussed so far can serve only as qualitative measurements that offer excellent insights into the controlling factors in the soot incipient formation process. In order to gain greater understanding of the soot processes in flames, it has been necessary to undertake extensive chemical and physical measurements of temperature and velocity as well as laser extinction, scattering, and fluorescence in premixed flames [84, 85], Wolfhard–Parker burners [86–88], coannular burners [89–91], and opposed-jet diffusion flames [63, 64, 92, 93] (see Ref. [51]).

475

Environmental Combustion Considerations

TABLE 8.7 Flow Rates* for Flames in Fig. 8.20 [74] C2H4

Ar

O2

C7H8

Total

C/O

Flame 1

1.68

8.89

2.05

0

12.62

0.82

Flame 2

1.68

8.89

2.05

0.044

12.66

0.89

Flame 3

1.83

8.89

2.05

0

12.77

0.89

Flame 4

2.17

9.12

2.44

0

12.73

0.89

*

In liters per minute (STP).

Harris and Weiner [74,75] have contributed much to the study of soot particle history in premixed flames. They used laser light scattering and extinction measurements to determine particle size, number density, and volume fraction in experiments with flat flame burners stabilized by a water-cooled porous plug. Their studies showed that richer ethene flames produce more soot owing to a higher nucleation rate, increased surface growth in these richer flames is induced by the increased surface area available for growth, and depletion of growth species does not occur in these flames. Therefore, these investigators concluded that the final particle size reached, when surface growth has virtually stopped, is determined not by growth species depletion, but rather by a decrease in the reactivity of the particles. They further concluded that the results are applicable to other fuels because the post-flame gases of aliphatic fuels are all quite similar, acetylene and methane being the prominent hydrocarbons present. Thus the similarity in surface growth in aliphatic fuels is related to the dominant role that acetylene plays in these processes. Although the post-flame gases of aromatic fuels have about 100 times more PAH than the post-flame gases of aliphatic fuels [94], acetylene remains the principal hydrocarbon in the burned gases of benzene flames [95]. This finding, combined with the fact that growth rates in benzene and ethylene are nearly identical [86], suggests that the mechanism of growth in benzene flames is also dominated by acetylene. These conclusions lend great validity to the qualitative results, postulates, and correlation given previously for premixed flames [53,96]. As stated, this surface growth pattern has been found to be the same in diffusion flames as well. For the flames described in Table 8.7, Harris and Weiner [74] obtained the results shown in Fig. 8.20, where the increase of soot volume fraction is plotted as a function of time. Thus, temperature measurements revealed that Flame 3 has the lowest temperature, Flames 2 and 4 are of equal, and somewhat higher temperature, and Flame 1 has the highest temperature. These results supply quantitative proof that in premixed flames the tendency to soot decreases with increasing flame temperature. Also of importance is the fact that Flame 2, which has toluene as a fuel constituent and the same equivalence ratio as that of Flame 4 for pure ethene, gives the same soot volume fraction as Flame 4

476

Combustion

(Fig. 8.20). A larger initial soot volume fraction indicates a larger initial incipient particle formation. The particle number density decreases with time due to coagulation; however, fuels that have a larger initial number density (incipient particle formation) have the largest final number density. Similar results for particle histories were found in opposed-jet diffusion flames [63, 64]. These quantitative results support the use of the qualitative critical sooting equivalence ratio results obtained on Bunsen flames as an excellent means for evaluating the sooting tendency of fuels under premixed conditions. One of the earliest detailed diagnostic efforts on sooting of diffusion flames was that of Wagner et al. [86–88], who made laser scattering and extinction measurements, profile determinations of velocity by LDV, and temperature measurements by thermocouples on a Wolfhard–Parker burner using ethene as the fuel. Their results show quite clearly that soot particles are generated near the reaction zone and are convected farther toward the center of the fuel stream as they travel up the flame. The particle number densities and generation rates decline with distance from the flame zone. The soot formation rate appeared to

Volume fraction (cm3 - soot/cm3 - flame)

106

107 Flame 1 Flame 2 Flame 3 Flame 4

108

0

5

10

15

Time (ms)

FIGURE 8.20 Soot volume fraction profiles for premixed flames. See Table 8.7 (from Harris and Weiner [74]).

477

Environmental Combustion Considerations

peak at values about 2–3 mm on the fuel side of the reaction zone. The particle number density decreases by coagulation and gradually levels off. This process leads to an increase in particle size and a simultaneous increase in soot volume fraction as the particles absorb hydrocarbon species. Thus the process is very similar to that found by Harris and Weiner in premixed flames. Smyth et al. [60] drew essentially the same conclusions in characterizing a sooting methane Wolfhard–Parker burner by detailed species profiles. Santoro et al. [89,90] performed some early extensive quantitative studies of coannular diffusion flames. They found that a striking feature of the coannular laminar diffusion flame is the existence of a roughly toroidal zone near the base of the flame where intense nucleation coexists with intense agglomeration to produce moderate-size particles. The soot formed in this region is convected upward along streamlines to form an annular region in which further gas-to-particle conversion occurs at moderate rates. Indeed, it is because of the intense particle formation in this toroidal region that the smoke height is identified as that point at which “wings” appear in the laminar diffusion flame [70]. The results of Santoro et al. [90], depicted in Fig. 8.21, also indicate that the 2000 105

Temperature (K)

107 1000

Qvv (cm1 sr1)

106

1500

108 Scattered light Temperature

500

10

5

Burner lip 0 Radial position (mm)

109 5

10

FIGURE 8.21 Comparison of the radial profiles for scattering cross-section Qvv and temperature (uncorrected) as a function of radial position for a coannular ethane diffusion flame (from Santoro et al. [85]).

478

Combustion

major soot formation regime is 2–3 mm on the fuel side of the flame, whose position is specified by the maximum temperature. In support of the concept postulated previously, one notes that the point of maximum particle density occurs at about 1600 K. The fact that particles exist at other temperatures along the analysis profile in Fig. 8.21 is attributable to thermophoresis. Furthermore, fluorescence measurements aimed at the detection of PAHs showed that the maxima clearly precede the maxima of soot volume fraction. No apparent difference in fluorescence levels was observed in experiments with the butene and benzene fuels, in contrast with the findings of other investigators [86]; thus, PAHs apparently have the same role in diffusion flames for both aliphatic and aromatic fuels. PAHs may, in fact, be involved in the soot nucleation stage (as has often been hypothesized and will be discussed next); such a finding, together with the relative constancy of the temperature at the soot onset, would suggest that, even though the extent of fuel to soot conversion may vary significantly from fuel to fuel, a common soot formation mechanism exists for all fuels. Kent and Wagner [97] made some very interesting quantitative measurements of overall axial soot concentrations as well as soot, flame zone, and centerline temperatures of coannular flames that have great significance. These results for ethene at different flow rates are shown in Fig. 8.22. The data indicate that flames emit smoke when the soot temperature in the oxidation zone falls below 1300 K. As the temperature of the soot decreases, the flow proceeds downstream because of cooling and radiation losses due to the soot mass formed interior to the flame zone. Thus the flow rates indicated in Fig. 8.22 for 3.4 and 5.7 ml/s would have flames below the smoke height, and the actual smoke height would exist for a volumetric flow rate between 5.7 and 6.1 ml/s. Establishing that smoke is emitted when burnout ceases as the soot temperature drops below 1300 K, regardless of other conditions, is a very significant and practical result. Thus, if incipient particle formation occurs in the 1650 K range under flame conditions and particle burnout ceases below about 1300 K range, a soot formation window exists for combustion processes. If one wishes to prevent soot from emanating from a combustion system operating in normal ranges, the temperature must be kept below about 1650 K. If this procedure is not possible, the exhaust must be kept lean and above 1300 K. With this understanding, one can visualize the “Catch-22” problem facing diesel engine operators. When diesel engines are operated lean to reduce the thermal NOx the manifold exhaust temperature becomes too low to burn the soot particles that will form; but when they are operated less lean to burn the soot, the NOx increases.

6. Chemical Mechanisms of Soot Formation The quantitative and qualitative results presented in the previous sections confirm some original arguments that the chemistry of fuel pyrolysis, and hence fuel structure, plays an important, and possibly dominant, role in sooting

479

Environmental Combustion Considerations

2100

Maximum

1700 Soot

Temperature (°K)

1300 Centerline 900 1.5 500 1.0 X 0.5

0 0

0.2

0.4

0.6

0.8

10

12

0

Z FIGURE 8.22 Axial distributions of relative soot yield x; maximum, centerline, and soot temperatures for ethene at flow rates around soot point in coaxial diffusion flames. Flow rates (ml/s): 䉫 3.4,  5.2, 䊊 6.1, Δ 7.2, 䊐 10.9 (from Kent and Wagner [92]).

diffusion flames. The identical chemical character of the soot formed from diffusion or premixed controlled systems seems to suggest a commonality in the chemical mechanisms for soot formation. Apparently, there is an underlying fuel-independent general mechanism that is modified only with respect to alternative routes to intermediates. These routes are simply affected by the combustion system temperature and the general characteristics of the initial fuel. Essentially, this concept implies that the relative propensity of one fuel to soot compared to another is primarily determined by differences in the initial rate of formation of the first and second ring structures; moreover, while the mechanisms controlling the growth of large condensed-ring aromatics, soot nucleation, soot growth, etc., remain essentially unchanged, the growth steps in large aromatic structures that lead to soot nucleation are significantly faster than the initial-ring formation [98]. Thus the formation of the initial aromatic rings controls the rate of incipient soot formation. As is well established now, the incipient soot formation particle concentration determines the soot volume fraction, that is, the total amount of soot formed. In support of this general contention that initial-ring formation is the soot-controlling step are the results

480

Combustion

correlating the sooting tendencies of several fuels as measured by time-offlight mass spectrometry coupled to a shock tube and chemical sampling in coannular normal and inverse diffusion flames [98]. These cases revealed the presence of allene, which was established kinetically to form benzene directly through a C3 route [99]. Miller and Melius [99a] also have formulated a chemical kinetic model of soot formation in which they argue that the first aromatic ring is most likely formed by reaction of two propynyl (C3H3) radicals; correlations [62] of experimental results on the propensity of C3 hydrocarbon fuels to soot appear to support this route. To explain the underlying principles of molecular growth to soot formation mechanisms related to ions [100], ring growth [101, 102], polyacetylene chains [103], Diels–Alder reactions, and neutral radicals [104] have been proposed [105]. However, the experimental results discussed on sooting tendencies of fuels under various flame configurations can be explained very well by a generalized mechanism based on soot precursor growth through a system dependent upon neutral radicals. The detailed modeling of soot formation in the shock tube pyrolysis of acetylene [106] and other fuels [107] provides the central basis for the fuelindependent general mechanisms suggested here. It must be noted, as well, that a large body of work by Howard et al. [108, 109] on premixed flames with regard to formation of aromatic species provides direct tests of the proposed mechanisms and are key to understanding and modeling soot formation. The dominant route, as proposed by Frenklach et al. [106], is the focus of this development. Indeed, this mechanism (Fig. 8.23) would occur mostly at high temperatures [106] and serves as the preferred route in premixed flames where the original fuel structure breaks down to acetylene and soot is formed in the high-temperature post-flame zone. Shock tube results by Colket [110] strongly suggest that formation of benzene from acetylene pyrolysis is largely attributable to acetylene addition to the n-butadienyl radical (C4H5), not C4H3. Thus, one could conclude that in diffusion flames where soot begins forming about 1600 K, the chemical route is alternate route A in Fig. 8.23. Colket’s results [110] for vinylacetylene in the same temperature regime indicate that the main route to benzene is through vinyl radical addition to the vinylacetylene as specified in Fig. 8.23 by alternate route B. These three paths offer excellent insight into the sooting tendencies measured in premixed, coflowing, and counterflowing diffusion flames and shock tubes. The fastest and most abundant route to the formation of the appropriate intermediates for the growth of the first aromatic ring determines the incipient particle formation rate and the relative propensity of a fuel to soot. By examining this fuel propensity to soot, it becomes evident that C4 hydrocarbons have a greater propensity to soot because they most readily form the butadienyl radical [103]. This hypothesis certainly explains the very early results of Schalla et al. [57] for the n-olefin series in which butene had the smallest smoke height. Temperature control was not necessary in Schalla’s tests with olefins because

H H

C

C

1, 3 Butadienyl radical

Vinyl radical H

H C

H H

C2H2

C

C2H2

H

Vinylacetylene H

H C

C

H

C C

C

C2H3

H

H

H C

C C

C

C4H3

H

C4H4

H

2 H3

C C2H2



H  C2 2

ALT RTE B

C

C

H

C4H5 ALT RTE A

H

C

H

(H)

Vinylacetylene radical H (H2)

Phenyl radical H C

C

Phenylacetylene

H

H

C C H H(H2)

C C

H

H

C

C

C

C H

Linear C6H5

ALT RTE C Allene

Methylacetylene

CH2CCH2

HC C CH3

C3H4

C H

H

C2H2

C

C

C

Cyclic C6H5

H

H C

C

C

C2H2 (H)

H

H C

Environmental Combustion Considerations

Acetylene

C3H4

C C H

General mechanisms for soot formation.

481

FIGURE 8.23

H

482

Combustion

all olefins have the same C/H ratio and hence near identical flame temperature histories. Butene, then, soots more than acetylene because it creates the butadienyl radical more readily. And butadiene soots more than either butene or acetylene, again because it more readily forms the butadienyl radical. The fact that most alkylated benzenes show the same tendency to soot is also consistent with a mechanism that requires the presence of phenyl radicals, concentrations of acetylene that arise from the pyrolysis of the ring, and the formation of a fused-ring structure. As mentioned, acetylene is a major pyrolysis product of benzene and all alkylated aromatics. The observation that 1-methylnaphthalene is one of the most prolific sooting compounds is likely explained by the immediate presence of the naphthalene radical during pyrolysis (see Fig. 8.23). Sampling in inverse coannular diffusion flames [62] in which propene was the fuel has shown the presence of large quantities of allene. Schalla et al. [57] also have shown that propene is second to butene as the most prolific sooter of the n-olefins. Indeed, this result is consistent with the data for propene and allene in Ref. 72. Allene and its isomer methylacetylene exhibit what at first glance appears to be an unusually high tendency to soot. However, Wu and Kern [111] have shown that both pyrolyze relatively rapidly to form benzene. This pyrolysis step is represented as alternate route C in Fig. 8.23. Detailed examination of the chemical reactions in the general mechanism in Fig. 8.23 reveals the importance of the pyrolysis fragments of all the fuels considered, especially the significance of H atoms, vinyl radicals, and acetylene. In particular, it becomes apparent that H atom and acetylene concentrations are important because they regulate, by abstraction and addition, respectively, the rapidity with which the large soot precursors can grow. The presence of large concentrations of H atoms in fuel pyrolysis is due to the thermal decay of the alkyl radicals present. Since the unimolecular cleavage of these radicals is not fast until temperatures well above 1100 K are reached [112], it is quite possible that this kinetic step and H atom diffusion set the range of the critical temperature for incipient soot formation in diffusion flames. Since there is no radical diffusion in shock tube studies of soot formation, it is not surprising, as mentioned, that the temperature at which soot formation is first identified is somewhat higher than that for diffusion flames.

7. The Influence of Physical and Chemical Parameters on Soot Formation The evidence suggests that temperature is the major parameter governing the extent to which a given fuel will soot under a particular flame condition or combustion process. As emphasized earlier, increasing the temperature under premixed fuel–air conditions decreases soot production, whereas the opposite is true for diffusion flames. The main effect of varying pressure must be manifested through its effect on the system temperature. Fuel pyrolysis rates are a

Environmental Combustion Considerations

483

function of concentration, but the actual dependence on the fractional fuel conversion depends on the overall order of the pyrolysis process. Unfortunately, varying the pressure alters the structure of the diffusion flame and the diffusivity of the fuel. As discussed, a variation in the diffusivity can shorten the growth period of the incipient particles formed. The diffusivity varies inversely with the pressure. Regrettably, the particular effect of an additive is difficult to assess. A chosen diluent cannot lower the flame temperature without altering the effective thermal diffusivity of the fuel–diluent jet. Thus, in examining the possible effect of an additive it is imperative to consider how the measurements of the extent of sooting were made. With respect to physical parameters, the effect of temperature is clear. However, the data reported on pressure variation must be viewed with caution since, as stated, pressure variations cause variations in temperature, thermal diffusivity, flow velocity, and flame structure. It now seems evident that additives that alter the early concentrations of H atoms can have a recognizable effect on the soot output. Again, the extent of the effect depends on the type of flame, the specific fuel, and the addition. Perhaps the best example is the addition of modest amounts of oxygen to the fuel in a diffusion flame configuration. The addition of oxygen to tightly bound fuels such as ethene, ethyne, and benzene noticeably affects the sooting tendency of these fuels. Similarly, a modest amount of oxygen added to the paraffins tends to act as an inert diluent [65]. Chemical kinetic analyses of the pyrolysis rates of these fuels at modest temperatures confirm that the effect in the case of the tightly bound fuels is to rapidly increase the H atom concentration, whereas rapid creation of H atoms due to their thermal homolysis of paraffins is little affected by the presence of oxygen. The general effect of continuously increasing the oxygen concentration on the fuel side of ethene and propane in opposed-jet diffusion flame experiments supports these conclusions and gives an overall picture of the effect of oxygen addition [65]. In these experiments, as the fuel–oxygen ratio was increased, the oxygen concentration in the opposed jet was reduced to keep the flame temperature constant. The particular results are shown in Fig. 8.19. The higher the extinction coefficients, the greater are the soot volume fractions. Ethene’s propensity to soot rises with decreasing φ (small oxygen addition). As φ decreases further, a chemical reaction releases energy, altering the temperature profile and increasing the sooting tendency still further. At a given φ, oxidizer species become present in sufficient abundance that the soot precursors are attacked. Eventually, oxidizer attack exceeds the thermal effect, a maximum is reached, and the conditions approach those of a premixed flame. The soot formed then decreases with decreasing φ, vanishing at a value near the critical sooting equivalence ratio determined by the conventional methods discussed. The same trends hold for propane except that the oxygen initially acts as a diluent, lowering soot production until the reaction is sufficiently initiated to raise the overall thermal profile level.

484

Combustion

The results in Fig. 8.19 are significant in that, for values of φ less than the maximum shown, premixed flame conditions exist. Since the flammability limit lies between the critical sooting equivalence ratio and the maximum φ for sooting, it is possible to conclude that, under all premixed flame conditions, fuel structure plays no role in the sooting tendency. Also, when φ decreases near infinity, the sooting tendency rises for ethene and decreases not only for propane, but would also for n-butane and isobutane [65]. The fact that these opposing trends can be explained by a natural radical reaction mechanism tends to support the notion that such mechanisms control the soot formation rate. It is difficult to see how the trends could be explained on the basis of an ion mechanism. Furthermore, the wide variation of sooting tendency with φ indicates the difficulty of extracting fundamental information from studies of turbulent fuel jets. The presence of halogen additives substantially increases the tendency of all fuels to soot under diffusion flame conditions [69]. The presence of H atoms increases the soot pyrolysis rate because the abstraction reaction of H  RH is much faster than R  RH, where R is a hydrocarbon radical. Halogenated compounds added to fuels generate halogen atoms (X) at modest temperatures. The important point is that X  RH abstraction is faster than H  RH, so that the halogen functions as a homogeneous catalyst through the system X  RH → R  XH H  XH → H 2  X Sulfur trioxide is known to suppress soot in diffusion flames and increase soot in premixed flames. These opposite effects can be explained in the context of the section on sulfur oxidation (Section 2e). In diffusion flames, the slight suppression can be attributed to the reaction of H atoms via the step H  SO3 → OH  SO2 If this step occurs late in the pyrolysis process, the hydroxyl radicals that form could attack the soot precursors. Thermal diffusivity may also have an effect. In premixed flames the SO3 must dissociate into SO2, which removes H atoms by H  SO2  M → HSO2  M The reduction in H concentration leads to a decrease in OH and hence to an increase in soot production. As additives to reduce soot output in flames, metal and organometallic compounds, such as ferrocene, have attracted the attention of many investigators (see Ref. [113]). The effect in premixed flames is best described by Bonczyk [114], who reported that the efficiency with which a given metal affects soot production characteristics depends almost exclusively on temperature and the

Environmental Combustion Considerations

485

metal atom’s ionization potential. Thus the alkaline-earth metals Ba Sr Ca are the most effective. The metal additives reduce the soot particle size and volume fraction and increase the number density. Essentially, ion charge transfer reduces agglomeration, and, under the appropriate conditions, particle burnup is augmented. Indeed, the burnup effect is prevalent in diffusion flames since the smaller particles must pass through the flame front. The conditions describing the smoke height are pertinent. There is no clear evidence that metal additives affect the incipient particle formation in diffusion flames. The difficulty in understanding the effect of ferrocene in diffusion flames is compounded by the fact that it decomposes into cyclopendienyl, which readily forms soot.

F. STRATOSPHERIC OZONE The ozone balance in the stratosphere is determined through complex interactions of solar radiation, meteorological movements within the stratosphere, transport to and from the troposphere, and the concentration of species based on elements other than oxygen that enter the stratosphere by natural or artificial means (such as flight of aircraft). It is not difficult to see that ozone initially forms from the oxygen present in the air. Chapman [115] introduced the photochemical model of stratospheric ozone and suggested that the ozone mechanism depended on two photochemical and two chemical reactions: O2  hv → O  O

(8.149)

O  O 2  M → O3  M

(8.150)

O3  hv → O2  O

(8.151)

O  O3 → O 2  O 2

(8.152)

Reactions (8.149) and (8.150) are the reactions by which the ozone is generated. Reactions (8.151) and (8.152) establish the balance, which is the ozone concentration in the troposphere. If one adds reactions (8.151) and (8.152), one obtains the overall rate of destruction of the ozone: namely, O3  hv → O2  O

(8.151)

O  O3 → O 2  O 2

(8.152)

net 2O3  hv → 3O2

(8.153)

The rates of reactions (8.149)–(8.152) vary with altitude. The rate constants of reactions (8.149) and (8.151) are determined by the solar flux at a given

486

Combustion

altitude, and the rate constants of the other reactions are determined by the temperature at that altitude. However, precise solar data obtained from rocket experiments and better kinetic data for reactions (8.150)–(8.152), coupled with recent meteorological analysis, have shown that the Chapman model was seriously flawed. The concentrations predicted by the model were essentially too high. Something else was affecting the ozone.

1. The HOx Catalytic Cycle Hunt [116, 117] suggested that excited electronic states of O atoms and O2 might account for the discrepancy between the Chapman model and the measured meteorological ozone distributions. But he showed that reactions based on these excited species were too slow to account for the differences sought. Realizing that water could enter the stratosphere, Hunt considered the reactions of free radicals (H, HO, HOO) derived from water. Consistent with the shorthand developed for the oxides of nitrogen, these radicals are specified by the chemical formula HOx. The mechanism that Hunt postulated was predicated on the formation of hydroxyl radicals. The photolysis of ozone by ultraviolet radiation below 310 nm produces excited singlet oxygen atoms that react rapidly with water to form hydroxyl radicals: O3  hv → O2  O (1D)

(8.154)

O (1D)  H 2 O → 2OH

(8.155)

Only an excited singlet oxygen atom could react with water at stratospheric temperatures to form hydroxyl radicals. At these temperatures, singlet oxygen atoms could also react with hydrogen or methene to form OH. The OH reacts with O3 to produce hydroperoxy radicals HO2. Both HO and HO2 destroy ozone by an indirect reaction that sometimes involves O atoms: HO  O3 → HO2  O2

(8.156)

HO2  O3 → HO  O2  O2

(8.157)

HO  O → H  O2

(8.158)

HO2  O → HO  O2

(8.159)

H  O3 → HO  O2

(8.160)

There are numerous reactions of HO2 radicals possible in the stratosphere. The essential reactions for the discussion of the ozone balance are

Environmental Combustion Considerations

487

HO  O3 → HO2  O2

(8.156)

HO2  O3 → HO  O2  O2

(8.157)

net 2O3 → 3O2 The reaction sequence (8.156)–(8.157) is a catalytic chain for ozone destruction and contributes to the net destruction. However, even given the uncertainty possible in the rates of these reactions and the uncertainty of the air motions, this system could not explain the imbalance in the ozone throughout the stratosphere.

2. The NOx Catalytic Cycle In the late 1960s, direct observations of substantial amounts (3 ppb) of nitric acid vapor in the stratosphere were reported. Crutzen [118] reasoned that if HNO3 vapor is present in the stratosphere, it could be broken down to a degree to the active oxides of nitrogen NOx (NO and NO2) and that these oxides could form a catalytic cycle (or the destruction of the ozone). Johnston and Whitten [119] first realized that if this were so, then supersonic aircraft flying in the stratosphere could wreak harm to the ozone balance in the stratosphere. Much of what appears in this section is drawn from an excellent review by Johnston and Whitten [119]. The most pertinent of the possible NOx reactions in the atmosphere are NO  O3 → NO2  O2

(8.3)

NO2  O → NO  O2

(8.161)

NO2  hv → NO  O

(8.1)

whose rate constants are now quite well known. The reactions combine in two different competing cycles. The first is catalytic destructive: NO  O3 → NO2  O2

(8.3)

NO2  O → NO  O2

(8.161)

net O3  O → O2  O2 and the second, parallel cycle is essentially a “do-nothing” one: NO  O3 → NO2  O2

(8.3)

488

Combustion

NO2  hv → NO  O O  O 2  M → O3  M

(8.1) (8.150)

net no chemical change The rate of destruction of ozone with the oxides of nitrogen relative to the rate in “pure air” (Chapman model) is defined as the catalytic ratio, which may be expressed either in terms of the variables (NO2) and (O3) or (NO) and (O). These ratio expressions are β

rate of ozone destruction with NO x rate of ozone destruction in pure air

(8.162)

β  1  {k161 (NO2 )/k152 (O3 )}

(8.163)

β  1  {(k3 k161 /k152 j1 )(NO)}/{1  k161 (O)/j1}

(8.164)

Here, as throughout this book, the k’s are the specific rate constants of the chemical reactions, while the j’s are the specific rate constants of the photochemical reactions. At low elevations, where the oxygen atom concentration is low and the NO2 cycle is slow, another catalytic cycle derived from the oxides of nitrogen may be important: NO2  O3 → NO3  O2

(8.165)

NO3  hv → (visible, day) → NO  O2

(8.166)

NO  O3 → NO2  O2

(8.3)

net 2O3  hv → 3O2 (day) The radiation involved here is red light, which is abundant at all elevations. Reaction (8.165) permits another route at night (including the polar night), which converts a few percent of NO2 to N2O5: NO2  O3 → NO3  O2

(8.165)

NO2  NO3  M → N 2 O5  M (night)

(8.167)

The rate of reaction (8.165) is known accurately only at room temperature, and extrapolation to stratospheric temperature is uncertain; nevertheless, the extrapolated values indicate that the NO3 catalytic cycle [reactions (8.165) and (8.166)] destroys ozone faster than the NO2 cycle below 22 km and in the region where the temperature is at least 220 K. The nitric acid cycle is established by the reactions

489

Environmental Combustion Considerations

HO  NO2  M → HNO3  M

(8.168)

HNO3  hv → OH  NO2

(8.169)

HO  HNO3 → H 2 O  NO3

(8.170)

The steady-state ratio of nitrogen dioxide concentration to nitric acid can be readily found to be [(NO2 )/(HNO3 )]ss  [k170 /k168 ]  [j169 /k168 (OH)]

(8.171)

For the best data available for the hydroxyl radical concentration and the rate constants, the ratio has the values 0.1 at 15 km,

1 at 25 km,

1 at 35 km

Thus it can be seen that nitric acid is a significant reservoir or sink for the oxides of nitrogen. In the lowest stratosphere, the nitric acid predominates over the NO2 and a major loss of NOx from the stratosphere occurs by diffusion of the acid into the troposphere where it is rained out. By using the HOx and NOx cycles just discussed and by assuming a NOx concentration of 4.2  109 molecules/cm3 distributed uniformly through the stratosphere, Johnston and Whitten [119] were able to make the most reasonable prediction of the ozone balance in the stratosphere. Measurements of the concentration of NOx in the stratosphere show a range of 28  109 molecules/cm3. It is possible to similarly estimate the effect of the various cycles upon ozone destruction. The results can be summarized as follows: between 15 and 20 km, the NO3 catalytic cycle dominates; between 20 and 40 km, the NO2 cycle dominates; between 40 and 45 km, the NO2, HOx, and Ox mechanisms are about equal; and above 45 km, the HOx reactions are the controlling reactions. It appears that between 15 and 35 km, the oxides of nitrogen are by far the most important agents for maintaining the natural ozone balance. Calculations show that the natural NOx should be about 4  109 molecules/cm3. The extent to which this concentration would be modified by anthropogenic sources such as supersonic aircraft determines the extent of the danger to the normal ozone balance. It must be stressed that this question is a complex one, since both concentration and distribution are involved (see Johnston and Whitten [119]).

3. The ClOx Catalytic Cycle Molina and Rowland [120] pointed out that fluorocarbons that diffuse into the stratosphere could also act as potential sinks for ozone. Cicerone et al.

490

Combustion

[121] show that the effect of these synthetic chemicals could last for decades. This possibly, a major source of atmospheric contamination arises by virtue of the widespread use of fluorocarbons as propellants and refrigerants. Approximately 80% of all fluorocarbons released to the atmosphere derive from these sources. There is no natural source. Eighty-five percent of all fluorocarbons are F11 (CCl3F) or F12 (CCl2F2). According to Molina and Rowland [120], these fluorocarbons are removed from the stratosphere by photolysis above altitudes of 25 km. The primary reactions are CCl3 F  hv → CCl2 F  Cl

(8.172)

CCl2 F  hv → CClF  Cl

(8.173)

Subsequent chemistry leads to release of additional chlorine, and for purposes of discussion, it is here assumed that all of the available chlorine is eventually liberated in the form of compounds such as HCl, ClO, ClO2, and Cl2. The catalytic chain for ozone that develops is Cl  O3 → ClO  O2

(8.174)

ClO  O → Cl  O2

(8.175)

net O3  O → O2  O2 Other reactions that are important in affecting the chain are OH  HO2 → H 2 O  O2

(8.176)

Cl  HO2 → HCl  O2

(8.177)

ClO  NO → Cl  NO2

(8.178)

ClO  O3 → ClO2  O2

(8.179)

Cl  CH 4 → HCl  CH3

(8.180)

Cl  NO2  M → ClNO2  M

(8.181)

ClNO2  O → ClNO  O2

(8.182)

ClNO2  hv → ClNO  O

(8.183)

ClO  NO2  M → ClONO2  M

(8.184)

The unique problem that arises here is that F11 and F12 are relatively inert chemically and have no natural sources or sinks, as CCl4 does. The lifetimes of these fluorocarbons are controlled by diffusion to the stratosphere, where photodissociation takes place as designated by reactions (8.172) and (8.173).

491

Environmental Combustion Considerations

TABLE 8.8 Residence Time of Halocarbons in the Troposphere* Halocarbon

Average residence time in years

Chloroform (CHCl3)

0.19

Methylene chloride (CHCl3)

0.30

Methyl chloride (CH3Cl)

0.37

1,1,1-Trichloroethane (CH3CCl3)

1.1

F12

330 or more

Carbon tetrachloride (CCl4)

330 or more

F11

1000 or more

*

Based on reaction with OH radicals.

The lifetimes of halogen species in the atmosphere are given in Ref. [122]. These values are reproduced in Table 8.8. The incredibly long lifetimes of F11 and F12 and their gradual diffusion into the stratosphere pose the problem. Even though use of these materials has virtually stopped today, their effects are likely to be felt for decades. Some recent results indicate, however, that the rate of reaction (8.179) may be much greater than initially thought. If so, the depletion of ClO by this route could reduce the effectiveness of this catalytic cycle in reducing the O3 concentration in the stratosphere.

PROBLEMS 1. Explain what is meant by atmospheric, prompt, and fuel NO. 2. Does prompt NO form in carbon monoxide–air flames? Why or why not? 3. Which is most sensitive to temperature—the formation of atmospheric, prompt, or fuel NO? 4. Order the tendency to soot of the following fuels under premixed combustion with air: hexadecane, ethyl benzene, cycloheptane, heptane, and heptene. 5. New freon agent compounds containing only fluorine atoms are being considered because fluorine atoms do not disturb the ozone layer in the manner as chlorine atoms. Can you explain why? 6. Order the following compounds in their tendency to soot under diffusion flame conditions: vinyl acetylene, ethene, phenyl acetylene, benzene, and acetylene.

REFERENCES 1. Bowman, C. T., Proc. Combust. Inst. 21, 859 (1992). 2. Leighton, P. A., “Photochemistry of Air Pollution,” Chap. 10. Academic Press, New York, 1961; Heicklen, J., “Atmospheric Chemistry.” Academic Press, New York, 1976.

492

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 17a. 17b. 17c. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 37a. 38. 39. 40. 41. 42. 43.

Combustion

Bulfalini, M., Environ. Sci. Technol. 8, 685 (1971). Blacet, F. R., Ind. Eng. Chem. 44, 1339 (1952). Dainton, F. S., and Irvin, K. J., Trans. Faraday Soc. 46, 374, 382 (1950). Miller, J. A., and Bowman, C. T., Prog. Energy Combust. Sci. 15, 287 (1989). Zeldovich, Y. B., Acta Physicochem. USSR 21, 557 (1946). Bowman, C. T., Prog. Energy Combust. Sci. 1, 33 (1975). Fenimore, C. P., Proc. Combust. Inst. 13, 373 (1971). Bowman, C. T., and Seery, D. V., “Emissions from Continuous Combustion Systems.” p. 123, Plenum, New York, 1972. Martenay, P. I., Combust. Sci. Technol. 1, 461 (1970). Bowman, C. T., Proc. Combust. Inst. 14, 729 (1973). Bachmeier, F., Eberius, K. H., and Just, T., Combust. Sci. Technol. 7, 77 (1973). Haynes, D. S., Iverach, D., and Kirov, N. Y., Proc. Combust. Inst. 15 (1975). Leonard, R. A., Plee, S. L., and Mellor, A. M., Combust. Sci. Technol. 14, 183 (1976). Eberius, K. H., and Just, T., “Atmospheric Pollution by Jet Engines,” AGARD Conf. Proc. AGARD-CP-125, p. 16–1 (1973). Hayhurst, A. N., and Vance, I. M., Combust. Flame 50, 41 (1983). Bozzelli, J.W., and Dean, A.M., Int. J. Chem. Kinet. 27, 1097 (1995). Konnov, A.A., Colson, G., and De Ruyck, J., Combust. Flame 121, 548 (2000). Rutar, T., Lee, J.C.Y., Dagaut, P., Malte, P.C., and Byrne, A.A., Proc. IMechE Vol. 221 PartA: J. Power and Energy, 376 (2007). Martin, G. B., and Berkau, E. E., AlChE Meet., San Francisco, 1971. Flagan, R. C., Galant, S., and Appleton, J. P., Combust. Flame 22, 299 (1974). Mulvihill, J. N., and Phillips, L. F., Proc. Combust. Inst. 15, 1113 (1975). Levy, A., Proc. Combust. Inst. 19, 1223 (1982). Fenimore, C. P., Combust. Flame 19, 289 (1972). deSoete, C. C., Proc. Combust. Inst. 15, 1093 (1975). Diehl, L. A., NASA Tech. Memo. NASA TM-X-2726 (1973). Hazard, H. R., J. Eng. Power Trans. ASME 96, 235 (1974). Hargraves, K. J. A., Harvey, R., Roper, F. G., and Smith, D., Proc. Combust. Inst. 18, 133 (1981). Courant, R. W., Merryman, E. L., and Levy, A., Int. Symp. Air Pollut., Health Energy Conserv., University of Massachusetts, (1981). Merryman, E. L., and Levy, A., Proc. Combust. Inst. 15, 1073 (1975). Cernansky, N. P., and Sawyer, R. F., Proc. Combust. Inst. 15 (1975). Allen, J. P., Combust. Flame 24, 133 (1975). Johnson, G. M., Smith, M. Y., and Mulcahy, M. F. R., Proc. Combust. Inst. 17, 647 (1979). Kesten, A. S., Combust. Sci. Technol. 6, 115 (1972). Bracco, F. V., Proc. Combust. Inst. 14, 831 (1973). Dryer, F. L., Mech. Aerosp. Sci. Rep., No. 1224. Princeton University, Princeton, NJ (1975). Myerson, A. L., Proc. Combust. Inst. 15, 1085 (1975). Lyon, R. K., U.S. Pat. 3,900,554; Int. J. Chem. Kinet. 8, 315 (1976). Miller, J. A., Branch, M. C., and Kee, R. J., Combust. Flame 43, 18 (1981). Perry, R. A., and Siebers, D. L., Nature (London) 324, 657 (1986). Levy, A., Merryman, E. L., and Ried, W. T., Environ. Sci. Technol. 4, 653 (1970). Cullis, C. F., and Mulcahy, M. F. R., Combust. Flame 18, 222 (1972). Johnson, G. M., Matthews, C. J., Smith, M. Y., and Williams, D. V., Combust. Flame 15, 211 (1970). Merryman, E. L., and Levy, A., Proc. Combust. Inst. 13, 427 (1971). Sachjan, G. A., Gershenzen, Y. M., and Naltandyan, A. B., Dokl. Akad. Nauk SSSR 175, 647 (1911). Muller, C. H., Schofield, K., Steinberg, M., and Broida, H. P., Proc. Combust. Inst. 17, 687 (1979).

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44. Myerson, A. L., Taylor, F. R., and Harst, P. L., J. Chem. Phys. 26, 1309 (1957). 45. Barrett, R. E., Hummell, J. D., and Reid, W. T., J. Eng. Power, Ser. A, Trans. ASME 88, 165 (1966). 46. Wendt, J. O. L., and Ekmann, J. M., Combust. Flame 25, 355 (1975). 47. deSoete, G. G., Inst. Fr. Pet., Fuel Rep., No. 23306 (1975). 48. Wendt, J. O. L., Morcomb, J. T., and Corley, T. L., Proc. Combust. Inst. 17, 671 (1979). 49. Palmer, H. B., and Cullis, H. F., “The Chemistry and Physics of Carbon,” Vol. 1. Dekker, New York, p. 205. 1965. 50. Haynes, B. S., and Wagner, H. G., Prog. Energy Combust. Sci. 7, 229 (1981). 51. Glassman, I., Proc. Combust. Inst. 22, 295 (1988). 52. Street, J. C., and Thomas, A., Fuel 34, 4 (1955). 53. Takahashi, F., and Glassman, I., Combust. Sci. Technol. 37, 1 (1984). 54. Miller, W. J., and Calcote, H. F., East. States Sect. Combust. Inst. Meet., East Hartford, CT, 1971; Calcote, H. F., Olson, D. B., and Kell, D. G., Energy Fuels 2, 494 (1988). 55. Wright, W. J., Proc. Combust. Inst. 12, 867 (1969). 56. Blazowski, W. S., Combust. Sci. Technol. 21, 87 (1980). 57. Schalla, R. L., Clark, T. P., and McDonald, G. E., Natl. Advis. Comm. Aeronaut. Rep., No. 1186 (1954). 58. Schalla, R. L., and Hibbard, R. R., Natl. Advis. Comm. Aeronaut. Rep., No. 1300, Chap. 9 (1959). 59. Wolfhard, H. G., and Parker, W. G., Proc. Phys. Soc. 2, A65 (1952). 60. Smyth, K. C., Miller, J. H., Dorfman, R. C., Mallard, W. G., and Santoro, R. J., Combust. Flame 62, 157 (1985). 61. Wu, K. T., and Essenhigh, R. H., Proc. Combust. Inst. 20, 1925 (1985). 62. Sidebotham, G., Ph.D. Thesis, Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, 1988. 63. Vandsburger, U., Kennedy, I., and Glassman, I., Combust. Sci. Technol. 39, 263 (1984). 64. Vandsburger, U., Kennedy, I., and Glassman, I., Proc. Combust. Inst. 20, 1105 (1985). 65. Hura, H. S., and Glassman, I., Proc. Combust. Inst. 22, 371 (1988). 66. Dryer, F. L., and Glassman, I., Prog. Astronaut. Aeronaut. 62, 225 (1978). 67. Glassman, I., AFOSR TR-79-1147 (1979). 68. Milliken, R. C., J. Phys. Chem. 66, 794 (1962). 69. Schug, K. P., Manheimer-Timnat, Y., Yaccarino, P., and Glassman, I., Combust. Sci. Technol. 22, 235 (1980). 70. Glassman, I., and Yaccarino, P., Proc. Combust. Inst. 18, 1175 (1981). 70a. Glassman, I., Proc. Combust. Inst. 27, 1589 (1998). 71. Gomez, A., Sidebotham, G., and Glassman, I., Combust. Flame 58, 45 (1984). 72. Glassman, I., “Combustion,” 2nd Ed. Academic Press, Orlando, FL, 1987. 73. Takahashi, F., Bonini, J., and Glassman, I., East. States Sect. Combust. Inst. Meet. Pap. No. 98 (1984). 74. Harris, S. J., and Weiner, A. M., Combust. Sci. Technol. 31, 155 (1983). 75. Harris, S. J., and Weiner, A. M., Proc. Combust. Inst. 20, 969 (1985). 76. Glassman, I., Brezinsky, K., Gomez, A., and Takahashi, F., in “Recent Advances in the Aerospace Sciences.” (C. Casci, ed.), p. 345. Plenum, New York, 1985. 77. Dobbins, R. A., Combust. Flame 130, 204 (2002). 78. Glassman, I., Nishida, O., and Sidebotham, G., “Soot Formation in Combustion.” p. 316. Springer- Verlag, Berlin, 1989. 79. Axelbaum, R. L., and Law, C. K., Proc. Combust. Inst. 23, 1517 (1990). 80. Rohsenow, W. M., and Choi, H., “Heat, Mass and Momentum Transfer.” Prentice-Hall, Englewood Cliffs, NJ, 1961. 81. Aksit, I. M., and Moss, J. B., Fuel 84, 239 (2005). 81a. Haggard, J. B., and Cochran, T. H., Combust. Sci. Technol. 5, 291 (1992). 82. Schug, K. P., Mannheimer-Timnat, Y., and Yaccarino, P., Combust. Sci. Technol. 22, 235 (1980).

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83. Pickett, L. M., and Siebers, D. L., Combust. Flame 138, 114 (2004). 84. D’Alessio, A., DiLorenzo, A., Borghese, A., Beretta, F., and Masi, S., Proc. Combust. Inst. 16, 695 (1977). 85. D’Alessio, A., in “Particulate Carbon Formation During Combustion.” (N. C. Siegla and G. W. Smith, eds.), p. 207. Plenum, New York, 1981. 86. Haynes, B. S., and Wagner, H. G., Ber. Bunsenges. Phys. Chem. 84, 499 (1980). 87. Kent, J. H., Jandes, H., and Wagner, H. G., Proc. Combust. Inst. 18, 1117 (1981). 88. Kent, J. H., and Wagner, H. G., Combust. Flame 47, 52 (1982). 89. Santoro, R. J., Semerjian, H. G., and Dobbins, R. A., Combust. Flame 51, 157 (1983). 90. Santoro, R. J., Yeh, T. T., and Semedian, H. G., Combust. Sci. Technol. 53, 89 (1987). 91. Gomez, A., Littman, M., and Glassman, I., Combust. Flame 70, 225 (1987). 92. Hura, H. S., and Glassman, I., Combust. Sci. Technol. 53, 1 (1987). 93. Hura, H. S., and Glassman, I., Proc. Combust. Inst. 22, 295 (1988). 94. Homann, K. H., and Wagner, H. G., Proc. Combust. Inst. 11, 371 (1967). 95. Bittner, J. D., and Howard, J. B., Proc. Combust. Inst. 18, 1105 (1981). 96. Takahashi, F., and Glassman, I., East. States Sect. Combust. Inst. Meet. Pap. No. 12 (1982). 97. Kent, J. H., and Wagner, H. G., Proc. Combust. Inst. 20, 1007 (1985). 98. Stein, S. E., and Kafafi, S. A., East. States Sect. Combust. Inst. Meet. Invited Pap. A (1987). Stein, S. E., J. of Phys. Chem. 82, 566 (1978). 99. Kern, R. D., Wu, C. H., Yong, J. N., Pamedemukkala, K. M., and Singh, H. J., Energy Fuels 2, 454 (1988). 99a. Miller, J. A., and Melius, C. F., Combust. Flame 91, 21 (1992). 100. Calcote, H. F., Combust. Flame 42, 215 (1981). 101. Gordon, A. S., Smith, S. R., and Nesby, J. R., Proc. Combust. Inst. 7, 317 (1959). 102. Crittenden, B. D., and Long, R., Combust. Flame 20, 239 (1973). 103. Glassman, I., AFOSR TR-79-1147 (1979). 104. Benson, S. W., Proc. Combust. Inst. 19, 1350 (1983). 105. Smyth, K. C., and Miller, J. H., Science 236, 1540 (1987). 106. Frenklach, M., Clary, D. W., Gardiner, W. C. Jr., and Stein, S., Proc. Combust. Inst. 12, 889 (1985). 107. Frenklach, M., Clary, D. W., Yuan, T., Gardiner, W. C. Jr., and Stein, S. E., Combust. Sci. Technol. 50, 79 (1986). 108. Cole, J. A., Bittner, J. D., Longwell, J. P., and Howard, J. B., Combust. Flame 56, 51 (1984). 109. Westmoreland, P. R., Howard, J. B., and Longwell, J. P., Proc. Combust. Inst. 21, 773 (1988). 110. Colket, M. B., III, Proc. Combust. Inst. 21, 851 (1988). 111. Wu, C. H., and Kern, R. D., J. Phys. Chem. 91, 6291 (1987). 112. Benson, S. W., Proc. Combust. Inst. 21, 703 (1988). 113. Zhang, J., and Megaridis, C. M., Proc. Combust. Inst. 25, 593 (1994). 114. Bonczyk, P. A., Combust. Flame 87, 233 (1991). 115. Chapman, S., Philos. Mag. 10, 369 (1930). 116. Hunt, B. G., J. Atmos. Sci. 23, 88 (1965). 117. Hunt, B. G., J. Geophys. Res. 71, 1385 (1966). 118. Crutzen, P. J., R. Meteorol. Soc. Q. J. 96, 320 (1970). 119. Johnston, H., and Whitten, G., in Advances in Environmental Science and Technology, Vol. 4, J.N. Pitts Jr. and R.L. Metcalf (eds), John Wiley & Sons, New York, pp. 236–380, 1974. 120. Molina, M. I., and Rowland, F. S., Nature (London) 249, 810 (1974). 121. Cicerone, R. J., Stolarski, R. S., and Walters, S., Science 185, 1165 (1974). 122. Federal Task Force on Inadvertent Modification of the Stratosphere, “Fluorocarbons and the Environment.” Counc. Environ. Qual., Washington, DC, 1975.

Chapter 9

Combustion of Nonvolatile Fuels

A. CARBON CHAR, SOOT, AND METAL COMBUSTION The final stages of coal combustion produce a nonvolatile char, which must be consumed to obtain good combustion efficiencies. The combustion of this char—a factor that has not yet been considered—is essentially a surface burning process similar to that occurring when carbon graphite burns. Coal is a natural solid fuel that contains carbon, moisture, ash-generating minerals, and a large number of different hydrocarbons that volatilize when combustion is initiated. The volatiles in coal contribute a substantial amount to the overall energy release. But the volatiles burn rapidly compared to the solid carbonaceous char that remains after devolatilization. It is the surface burning rate of this remaining nonvolatile solid carbonaceous fuel that determines the efficiency of the coal combustion process. Similarly, the emission of soot from many practical devices, as well as from flames, is determined by the rate of oxidation of these carbonaceous particles as they pass through a flame zone and into the post-combustion gases. As mentioned in the previous chapter, the soot that penetrates the reaction zone of a co-annular diffusion flame normally burns if the temperatures remain above 1300 K. This soot combustion process takes place by surface oxidation. Heterogeneous surface burning also arises in the combustion of many metals. Since the energy release in combustion of metals is large, many metals are used as additives in solid propellants. Indeed, in the presence of high oxygen concentrations, metal containers, particularly aluminum, have been known to burn, thereby leading to serious conflagrations. Not all metals burn heterogeneously. The determination of which metals will burn in a heterogeneous combustion mode can be made from a knowledge of the thermodynamic and physical properties of the metal and its oxide product [1]. In the field of high-temperature combustion synthesis, metals have been reacted with nitrogen, both in the gaseous and liquid phases, to form refractory nitrides [2]. In most cases, this nitriding process is heterogeneous. 495

496

Combustion

B. METAL COMBUSTION THERMODYNAMICS 1. The Criterion for Vapor-Phase Combustion What is unique about metal particles burning in oxygen is that the flame temperature developed is a specific known value—the vaporization–dissociation or volatilization temperature of the metal oxide product. This temperature could be referred to as a boiling point. This interesting observation is attributable to the physical fact that the heat of vaporization–dissociation or decomposition of the metal oxide formed is greater than the heat available to raise the condensed state of the oxide above its boiling point. That is, if QR is the heat of reaco ) is the tion of the metal at the reference temperature 298 K and (HTo  H 298 vol enthalpy required to raise the product oxide to its volatilization temperature at the pressure of concern, then o )  ΔH ΔH vapdissoc QR  (HTo  H 298 avail

(9.1)

vol

where ΔHvap–dissoc is the heat of vaporization–dissociation of the metal oxide [3]. Note that QR is the negative of the heat of formation of the metal oxide. Equation (9.1) assumes conservatively that no vaporization or decomposition takes place between the reference temperature and the volatilization temperature. The developments in Chapter 6 show that in the steady state the temperature of the fuel particle undergoing combustion approaches its boiling point (saturation) temperature at the given pressure. Characteristically, it is a few degrees less. For a condensed-phase fuel to burn in the vapor phase, the flame temperature must be greater than the fuel saturation temperature so that the fuel will vaporize and diffuse toward the oxidizing atmosphere to react. For liquid hydrocarbon fuels, the droplet flame temperature is substantially higher than the fuel saturation temperature. However, many metals have very high saturation temperatures. Thus, for a metal to burn as a vapor, the oxide volatilization temperature must be greater than the temperature of the metal boiling point. This statement is known as Glassman’s criterion for the vapor-phase combustion of metals. The metals that will burn in the vapor phase in oxygen can then be determined by comparing the second and last columns of Table 9.1 for the appropriate oxide.

2. Thermodynamics of Metal–Oxygen Systems Modern computational programs [4] and thermodynamic tables [5] now make it possible to explicitly calculate metal–oxygen flame temperatures, thereby opening up a unique aspect of combustion thermodynamics that could be important in the consideration of metal as fuels and as reactants in combustion synthesis.

497

Combustion of Nonvolatile Fuels

TABLE 9.1 Various Properties of Metal Oxides and Nitrides Metallic compound

a,b Tvol

(K)

a,b ΔHvol (kJ/mol)

 ΔHvol (kJ/mol)

ΔHf,o298 (kJ/mol)

620 (620)

740 (730)

318

2791

1676

2791

251

4139

o ) (HoTvol  H298

Metal Tbc(K)

AlN

2710 (2710)

Al2O3

4000

BN

4010 (2770)

840 (300)

950 (410)

B2O3

2340

360

640

1272

4139

BeO

4200

740

1060

608

2741

Cr2O3

3280

1160

1700

1135

2952

FeO

3400

610

830

272

3133

Li2O

2710

400

680

599

1620

MgO

3430

670

920

601

1366

Si3N4

3350 (2150)

2120 (870)

2420 (1160)

745

3505

TiN

3540 (3450)

700 (450)

950 (710)

338

3631

Ti3O5

4000

1890

2970

2459

3631

ZrO2

4280

920

1320

1097

4703

1860

2550

Note: Tvol, Volatilization temperature (or stoichiometric adiabatic combustion temperature creating compound under ambient conditions T  298 K, P  1 atm); Tb, Metal boiling point at 1 atm; Te, Decomposition temperature (see text). a Values reported are rounded to the nearest integer. b Value in parentheses are (or are based on) decomposition temperatures, Td, or enthalpies of decomposition, ΔHd. c Values from JANAF tables.

As early as 1953, Brewer [6] elaborated on the fact that an equilibrium thermodynamic boiling point is difficult to define for many metal oxides. Indeed, the experimental evidence demonstrated that many metal oxides volatilize exclusively by decomposition to gaseous suboxides and elements such as O2, or by dissociation to species such as O. Few actually formed gaseous molecules of the original metallic oxide. Subsequent investigators [1, 7] recognized the importance of dissociation in metal combustion systems. But despite the understanding of the thermodynamic definition of boiling point implied by this recognition, the term was used to describe the vaporization–decomposition process of metal oxides. The JANAF tables [5] recognize the difficulty of defining a true boiling point for metallic oxides. In a few cases, a “decomposition temperature” is defined for a particular reaction equilibrium. For MgO, for example, a “decomposition or boiling temperature (3533 K)” is defined

498

Combustion

at 1 atm, where the free energy change of the reaction MgO(l) → MgO(g) approaches zero. Similarly, a heat of vaporization is defined as the difference between ΔH f,o3533 for MgO(g) and MgO(l), where ΔH fo is the standard state heat of formation [5]. This definition, however, is not acceptable in the case of many metal oxides, including MgO. The term “boiling temperature” would be correct for the Mg–O2 system if MgO(l) were vaporized congruently to MgO(g). But as the JANAF tables note, “vaporization near 2000 K is primarily to the elements Mg(g) and presumably O(g) and O2(g) rather than to MgO(g)” [5]. The term “vaporization–decomposition” temperature would probably be more appropriate to describe the volatilization of metallic oxides. In fact, this temperature was later defined as a “volatilization temperature, exhibiting characteristics of a transition temperature or boiling point” [6]. Furthermore, in order to distinguish between the condition of complete decomposition of a nitride and the condition in which all the decomposed products are gases, the decomposition temperature and volatilization temperatures will be defined more explicitly. Von Grosse and Conway [7] were the first to realize that the evaluation of the adiabatic combustion temperatures for metal–oxygen systems was largely dependent on the metal oxide volatilization characteristics. Recognizing that many metallic oxides do not exert any appreciable vapor pressure until they reach very high temperatures, these investigators concluded that the enthalpy required for complete volatilization was large. Citing aluminum oxide as an example, von Grosse and Conway noted that “only about 140 kcal/mol are required to heat aluminum oxide to its boiling point … The heat amounting to the difference between 400.2 (the enthalpy of reaction) and 140 kcal must be interpreted as being used to vaporize and decompose the aluminum oxide at its boiling point (aluminum oxide decomposes on vaporization). The combustion temperature is thus limited to the boiling point of the oxide [emphasis added]. This is the case in many metal–oxygen systems, but each reaction must be evaluated independently.” The current ability to readily calculate combustion temperatures has confirmed the concept of a “limiting temperature” [1, 7] in metal processes. The term limit temperature is now used as a formal statement that the enthalpy of combustion of the reactants at the stoichiometric ambient (298 K) condition is insufficient to volatilize all the condensed-phase metallic oxide formed; that is, the metallic oxide undergoes a phase change. This concept of a limit temperature can be applied to metal–nitrogen systems as well [8]. Shown in Fig. 9.1 are the adiabatic combustion temperatures at 1 atm, calculated for the oxidation and nitriding of many metals at 298 K as a function of the equivalence ratio φ. Of course, a value of φ greater or less than 1 signifies an excess of one or the other of the reactants. Note that the temperature variation over a wide range (0.5–1.5) of equivalence ratios varies by only a few percent for many metal–oxygen/nitrogen systems. Boron is an exception because, at the stoichiometric value with oxygen, no condensed phases exist in the combustion

499

Combustion of Nonvolatile Fuels

4500 Zr–O2 AI–O2

Temperature (K)

4000 B–O2

Ti–N2

3500 Mg–O2 3000

Cr–O2 B–N2 AI–N2

2500 Si–N2 2000 0.5

0.7

0.8

1.0

1.2

1.3

1.5

Equivalence ratio (φ) FIGURE 9.1 Adiabatic combustion temperature of various metal–oxygen/nitrogen systems as a function of equivalence ratio φ. Initial conditions 298 K and 1 atm.

products. Off stoichiometric for other metal–oxygen/nitrogen systems, a temperature is reached that will vary minimally as stoichiometric is approached. The only reason there is a variation with equivalence ratio is that the equilibria of the decomposed metal oxide or nitride species are influenced by an excess of one of the reactants. Of course, it is possible to be so far off stoichiometric that the temperature drops appreciably to a value indicative of transition temperatures for other condensed-phase products formed. The point is that the very weak dependence of the combustion temperature over a wide range of equivalence ratios near stoichiometric further confirms the reasoning that the adiabatic combustion temperature in pure oxygen and nitrogen is limited due to the volatilization of the condensed-phase product formed. It would be possible, of course, to exceed this limiting temperature for a stoichiometric metal–oxygen/nitrogen system if the reactants were initially at very high temperatures. The equilibrium combustion temperatures and compositions of many metal–oxygen/nitrogen systems were calculated by assigning various values of the total enthalpy, H To,, of these reactants at a given, fixed total pressure [8]. One should recognize that this procedure is analogous to varying the total enthalpy or the enthalpy of formation of the product metal oxide or nitride. Performing these calculations for stoichiometric proportions of metal and oxidizer provides a means for the determination of the enthalpies needed to completely decompose and/or volatilize the metal oxide or nitride in question. Shown in Figs. 9.2–9.9 are the results for most of the systems designated by the product compounds listed in Table 9.1. An input enthalpy of zero is a condition equivalent to one in which the reactants enter at the ambient (298 K)

500

Combustion

1.00

5000

0.60

AI2O3(I) O

3000

0.40 AI

AIO

2000

Mole fraction

Temperature (K)

0.80

Temperature

4000

0.20 AI2O

1000 15

O2 10

5

0

5

0.00 15

10

Assigned enthalpy (kJ/g) FIGURE 9.2 Equilibrium product composition and adiabatic combustion temperature for stoichiometric Al and O2 at 298 K and 1 atm as a function of assigned enthalpy. An assigned enthalpy of zero corresponds to the true ambient condition. (l) specifies a liquid product, (s) specifies a solid product and all other products are gas.

1.00

3000 Temperature

0.80

2400

Al

AIN(s)

0.60

0.40

2100

Mole fraction

Temperature (K)

2700

N2 1800

0.20 Al(I)

1500 6 FIGURE 9.3

3

0 3 6 Assigned enthalpy (kJ/g)

9

0.00 12

Plot similar to Fig. 9.2 for the stoichiometric Al–N2 system.

state. Examining Fig. 9.2 for the Al–O2 system, one notes that the product composition shows a 0.216 mole fraction of Al2O3 liquid at this initial reactant state. As the input enthalpy increases, the temperature remains the same, but the amount of Al2O3 liquid decreases. When the assigned enthalpy reaches 8.56 kJ/g reactants, condensed-phase Al2O3 no longer exists and any further

501

Combustion of Nonvolatile Fuels

4500 BN(s)

Temperature

0.80

B(l) B

3700

3300

N2

2900

0.60

0.40

Mole fraction

Temperature (K)

4100

1.00

0.20 BN

2500 10

0

10

0.00 30

20

Assigned enthalpy (kJ/g) FIGURE 9.4 Plot similar to Fig. 9.2 for the stoichiometric B–N2 system.

4000

1.00 B2O3(I)

Temperature B2O3

0.80

0.60

3200 BO 2800

0.40 BO2

2400 O 2000 15

0.20

O2

0 10 5 Assigned enthalpy (kJ/g)

Mole fraction

Temperature (K)

3600

5

0.00

FIGURE 9.5 Plot similar to Fig. 9.2 for the stoichiometric B–O2 system.

increase in the total assigned enthalpy raises the temperature. It is also possible to withdraw enthalpy from the system. Again, the temperature does not change, but the amount of Al2O3 liquid increases. When an enthalpy of 9.68 kJ/g reactants is withdrawn, the temperature begins to drop, and the product composition contains only condensed-phase Al2O3. These results verify that a “pseudo-phase change” exists and that the limiting combustion equilibrium temperature has characteristics “similar to a boiling point.” An enthalpy

502

Combustion

1.00

5000 BeO(l)

0.80 Temperature

4200

0.60 Be O

3800

0.40

3400

Mole fraction

Temperature (K)

4600

0.20 BeO

3000 16

Be2O 8

0

8

16

0.00 32

24

Assigned enthalpy (kJ/g) FIGURE 9.6

Plot similar to Fig. 9.2 for the stoichiometric Be–O2 system.

1.00

4200

Ti3O5(I)

0.80 Temperature

4000

0.60 TiO

3900

0.40 TiO2

3800

O 0.20

3700 3600 8

Mole fraction

Temperature (K)

4100

O2 6

4

2

Ti 0

2

4

6

0.00

Assigned enthalpy (kJ/g) FIGURE 9.7

Plot similar to Fig. 9.2 for the stoichiometric Ti–N2 system.

of volatilization can thus be calculated for Al2O3 (MW 102) to be {[8.56  (9.68)]102}  1860 kJ/mol. Obviously, this enthalpy of volatilization/decomposition holds only for the stoichiometric condition, or more precisely, when the ratio of metal to oxygen (nitrogen) reactants is equal to that of the oxide (nitride) in question. This value of 1860 kJ/mol plus the enthalpy 688 kJ/mol needed to raise Al2O3 liquid to the limiting temperature is clearly greater than 1676 kJ/mol, the overall enthalpy of reaction or the enthalpy of formation of

503

Combustion of Nonvolatile Fuels

1.00

4000

0.80 Temperature

3500

0.60 Ti 0.40 3000

N2

Mole fraction

Temperature (K)

TiN(l)

0.20 Ti(l) 2500 2

0

2

4

6

8

0.00 12

10

Assigned enthalpy (kJ/g) FIGURE 9.8 Plot similar to Fig. 9.2 for the stoichiometric Ti–O2 system.

1.00

4000

3500

0.80 Temperature 0.60 Mg 0.40

3000

Mole fraction

Temperature (K)

MgO(I)

O 0.20 MgO, O2 2500 12

8

4 0 4 Assigned enthalpy (kJ/g)

8

0.00 12

FIGURE 9.9 Plot similar to Fig. 9.2 for the stoichiometric Mg–O2 system.

the metal oxide at the ambient condition (see Table 9.1). This example and the results shown in Figs. 9.2–9.9 appear to verify that limiting combustion temperatures are reached for many metal–oxygen/nitrogen systems. The approach has been adopted for alloys as well [9]. It is also worthwhile to point out that the gaseous species listed in Fig. 9.2 have all been identified in the combustion of aluminum. However, it is to be noted that in the case of aluminum, and perhaps other metals, that the chemical structure of the condensed phase product

504

Combustion

is not found to exist in the phase. The actual physics of the chemical nucleation process to form condensed phase Al2O3 from the gaseous species is still not understood. It is necessary to distinguish between the limiting decomposition temperature and the limiting volatilization temperature as well as their corresponding enthalpies of decomposition and volatilization. For example, depending on the system and system pressure, a decomposing refractory product may produce a species in a condensed phase. There is a temperature associated with this condition designated as the limiting decomposition temperature. As the assigned enthalpy is increased past the value that produced the limiting decomposition temperature, the condensed-phase product of decomposition volatilizes further. The temperature increases gradually with respect to this assigned enthalpy increase until the condensed-phase product species is completely vaporized, at which point the temperature rises sharply. The point of this sharp rise is designated as the volatilization temperature. At 1 atm, none of the oxide systems exhibits a condensed-phase decomposition product so that the decomposition and volatilization temperatures, Td and Tvol, are the same. These two limiting temperatures and their corresponding enthalpies are readily seen in Fig. 9.7 for the Ti–N2 system at 1 atm. This consideration could be particularly important in various combustion synthesis approaches. A volatilization temperature for the B–O2 system can be defined only if sufficient enthalpy is withdrawn to allow for B2O3(l) (see Fig. 9.5). This value of 2340 K corresponds exactly to the “boiling point” (2340 K) reported in the JANAF tables as the temperature at which the fugacity is 1 atm for B2O3(l) → B2O3(g). Since B2O3(l) vaporizes congruently to B2O3(g), the boiling point reported for this case should equal the calculated volatilization temperature. The JANAF tables specify a volatilization temperature of a condensedphase material to be where the standard-state free energy ΔGfo approaches zero for a given equilibrium reaction, that is, MxOy(l)  MxOy(g). One can obtain a heat of vaporization for materials such as Li2O(l), FeO(l), BeO(l), and MgO(l), which also exist in the gas phase, by the differences in the ΔH fo of the condensed and gas phases at this volatilization temperature. This type of thermodynamic calculation attempts to specify a true equilibrium thermodynamic volatilization temperature and enthalpy of volatilization at 1 atm. Values determined in this manner would not correspond to those calculated by the approach described simply because the procedure discussed takes into account the fact that some of the condensed-phase species dissociate upon volatilization. Examination of Fig. 9.4 for the B–N2 system reveals that BN decomposes into gaseous nitrogen and liquid boron. Since these elements are in their standard states at 1 atm and the decomposition temperature, the ΔH do must equal the enthalpy of formation ΔH fo of the BN at the decomposition temperature. Indeed, the ΔH do (300 kJ/mol) calculated by the means described agrees with the value of ΔH f,oT (300 kJ/mol) given in the JANAF tables, as it should. The d same condition holds for the Si–N2 system.

Combustion of Nonvolatile Fuels

505

As noted in Fig. 9.8, over the range of assigned enthalpies that define the limiting decomposition temperature, the major TiN decomposition products are Ti(l), Ti(g), and N2. Since both Ti(g) and Ti(l) exist, the products of decomposition are not in their standard states. For TiN, then, unlike BN, the enthalpy of decomposition ΔH do will not equal the enthalpy of formation at the decomposition temperature. When the assigned enthalpy is increased to 9.9 kJ/g reactants for the Ti–N2 system, the decomposition products are only gases ([Ti(g) and N2]. This condition specifies a volatilization temperature of 3540 K and a partial Ti(g) pressure of 0.666 atm. At 0.666 atm, the vaporization temperature for titanium has been found to be 3530 K. Indeed, neglecting dissociation, the values should be the same because the vaporization of Ti becomes the limiting condition. The enthalpy of volatilization for TiN, as determined by the procedure described here, is then found to be 700 kJ/mol. A value for the enthalpy of volatilization can be estimated as the heat of formation of TiN at 3630 K [7], the vaporization temperature of Ti at 1 atm. This value is of the order of 690 kJ/mol. Since the enthalpy of vaporization of Ti is not a function of pressure or temperature, the agreement between the value of enthalpy of volatilization of TiN at 3540 K and the estimated value of the enthalpy of formation of TiN from the JANAF tables at 3630 K, where the value has been determined at 1 atm, is not surprising. The volatilization process of the metal oxides behaves like a “pseudoboiling point.” For the volatilization of liquid Al2O3 alone, Brewer and Searcy [10] clearly pointed out that AlO(g) is the principal species, so that Al2O3 → 2AlO  O is the principal reaction of concern. From Fig. 9.2, one sees clearly that as enthalpy is increased in the stoichiometric Al–O2 system, the equilibrium shifts to favor more AlO(g) and O(g) and less Al2O3(l). As in the case of Al2O3, such dissociated species as O are important in the volatilization process of many metal oxides. Dissociation is also evident with the nitrides. Therefore, if the pressure is increased, as dictated by Le Chatelier’s principle, less dissociation occurs; thus, a smaller enthalpy of volatilization or decomposition and a higher adiabatic combustion temperature are found for most of the metal– oxygen and nitrogen systems examined. Fig. 9.10 depicts the temperature variation of the Al–O2 system over a tenfold variation in pressure. Indeed, if the total pressure is increased from 1 to 10 atm in the stoichiometric Al–O2 system, the heat of volatilization decreases by approximately 7% and the temperature increases by 15%. Similarly, there is an approximate 4% decrease in the volatilization enthalpy and a 17% increase in the temperature for the Mg–O2 system. With respect to the compositions shown in Figs. 9.2–9.9, the variations in the species composition can be explained by the combination of temperature and pressure effects, not just by pressure alone. The volatilization or pseudo-boiling point temperature and the corresponding enthalpies of decomposition and volatilization of all the systems examined as a function of total pressure are reported in Table 9.2. The variation in the enthalpy of volatilization as a function of pressure is generally less than 7%. o with pressure. The Ti–O2 system (Fig. 9.8) shows an anomalous rise in ΔH vol

506

Combustion

5100

1.00 1 atm 10 atm

Al2O3(1)

0.80 Temperature 0.60

4100

Temperature O

0.40

Mole fraction

Temperature (K)

4600

AIO 3600 0.20 AI 3100 15

10

5

0

5

10

0.00 15

Assigned enthalpy (kJ/g) FIGURE 9.10 Comparison of the plot of adiabatic combustion temperature and product composition of the stoichiometric Al–O2 system at total pressure of 1 atm (Fig. 9.2) and that at 10 atm.

When the Al–N2 system is raised from 0.1 to 10 atm, the Al(g) product condenses and the species of decomposition are no longer in their standard states. When the assigned enthalpy is increased further so that the Al returns to the o gaseous state, an enthalpy of volatilization ΔH vol based on this new assigned o enthalpy can be defined. The values of ΔH vol specify these new values based on the condition that the elements are in their gaseous states at the pressure o for the metal–nitrogen and volatilization temperature. When values of ΔH vol systems are calculated for the condition in which the decomposition products o , unlike are completely in the gaseous state, the pressure variation of ΔH vol o ΔH d , is minimal. o of the Ti–O system rises The apparently anomalous result that the ΔH vol 2 with pressure is explained by examining the volatilization product compositions as a function of pressure. The following equilibria exist for this system: Ti3 O5 (1)  2 TiO2  TiO

(9.2)

TiO2  TiO  O

(9.3)

As the pressure is increased in the system, the equilibrium designated as reaction (9.2) shifts to Ti3O5(l) because the pressure effect overrides the increase in temperature with pressure that would make the shift to the gaseous products. Indeed, at a given assigned enthalpy, the Ti3O5(l) mole fraction increases as the total pressure increases. For the equilibrium designated as

b,c Tvol (K)

Metallic compound

b,c ΔHvol (kJ/mol)

0.1 atm

1 atm

10 atm

0.1 atm

1 atm

10 atm

AlN

2110 (2410)

2710 (2710)

3290 (3030)

630 (630)

620 (620)

620 (370)

Al2O3

3540

4000

4620

BN

3470 (2540)

4010 (2770)

4770 (3040)

830 (300)

840 (300)

850 (300)

B2O3

2080

2340

2670

370

360

360

BeO

3640

4200

4980

740

740

730

Cr2O3

2910

3280

3760

1180

1160

1140

FeO

2940

3400

4050

610

610

610

Li2O

2330

2710

3270

410

400

400

MgO

(s)

3430

4020

(s)

670

640

3350 (2150)

4000 (2370)

3540 (3450)

4250 (3960)

Si3N4 TiN

2880 (1960) (s)

1970

2120 (870) (s)

1860

1750

2120 (870)

2140 (860)

700 (460)

700 (370)

Ti3O5

3470

4000

4740

1800

1890

1930

ZrO2

3790

4280

4920

930

920

910

507

Note: (s), Solid form. a Temperatures are those obtained by the reacting systems creating the products under stoichiometric conditions for initial temperature of 298 K. b Values reported are rounded to nearest integer. c Values in parentheses are decomposition temperatures, Td or enthalpies of decomposition, ΔHd.

Combustion of Nonvolatile Fuels

TABLE 9.2 Temperatures and Heats of Volatilization of Various Oxide and Nitride Products at Various Pressuresa

508

Combustion

reaction (9.3), however, the increase of temperature with pressure overrides the shift that would occur due to pressure; so there is more dissociation, resulting o in a greater quantity of O atoms. This trend explains the increase in ΔH vol with pressure for the Ti–O2 system and for other systems that form the more complex stable oxides. Considering that dissociation occurs upon volatilization, the temperatures can be correlated extremely well on a ln P versus (1/Td,vol) plot, where P is the total system pressure and Td,vol is the volatilization or decomposition temperature, as the case dictates. Such a plot is shown in Fig. 9.11. Since the Clausius– Clapeyron relation for vapor pressure of pure substances shows an exponential dependence on temperature, Tvol was considered a pseudo-boiling point at the respective system pressure. For a substance that vaporizes congruently to its gaseous state, the slope of lines on a ln P versus (1/Tvol) plot represents the enthalpy of vaporization. Indeed, the enthalpy of vaporization calculated from the slope on a ln P versus (1/Tvol) plot for the B–O2 system (360 kJ/mol) agrees exactly with the value calculated by using the procedure outlined. Since the other metal–oxides/nitrides examined do not vaporize congruently to their gaseous state, it is quite apparent that enthalpies of dissociation play a role in determining the slope of the ln P versus (1/Tvol) plots. Table 9.1 and Fig. 9.11 also depict vaporization temperatures of the metals in each product composition and give a graphical representation of Glassman’s criterion. When Tvol (or Td, as the case dictates) of the refractory compound formed is greater than the vaporization temperature, Tb, of the metal reactant, small metal particles will vaporize during combustion and burn in the vapor phase. When the contra condition holds, much slower surface reactions will 100

Pressure (atm)

AIN(s)

10

ZrO2(l) TiN(l) Al

1

Zr

Al2O3(l)

0.1 0.18

0.22

0.26 1000 (K)/T

0.30

Ti

0.34

FIGURE 9.11 Vapor pressure–temperature relationships of various vaporization and combustion temperatures of various metal–oxygen/nitrogen systems as a function of total pressure, both plotted in the form log P versus 1/T.

509

Combustion of Nonvolatile Fuels

take place. This temperature condition could change with pressure; however, change is not likely to occur over a large pressure variation for combustion in pure oxygen [3]. Thus in pure oxygen, Al, Be, Cr, Li, Mg, and Ti fit the criterion for vapor-phase combustion while B and Zr do not. The temperatures for the criterion for vapor-phase combustion at 1 atm for Fe and Ti are close (about 300 K for Fe and 400 K for Ti); consequently, thermal losses from the flame front would make the actual flame temperature less than the volatilization temperature of FeO and Ti3O5, just to complicate matters. With regard to combustion synthesis processes, it would appear that, at least for nitride formation, heterogeneous surface reactions would dominate the synthesis procedure.

3. Thermodynamics of Metal–Air Systems In the metal reaction systems described in the preceding section, the gaseous atmosphere was either pure oxygen or pure nitrogen. Two questions now arise: Would a metal burning in air have a flame temperature equal to the vaporization–dissociation temperature of the metal oxide at the total pressure of the system? And would a temperature plateau exist over a range of equivalence ratios or over a range of assigned enthalpies? Figure 9.12 details the same type of stoichiometric calculations as shown in earlier figures, except that given amounts of inert (argon) are added to an aluminum–oxygen mixture [11]. In one case, 8.46 mol of argon is added to the stoichiometric amount of oxygen, and in another 2.82 mol. Argon was considered instead of nitrogen because the formation of gaseous nitrogen oxides

4500

Temperature (K)

4000

2Al  1.5O2 (1.0 atm) 1 2

3500

4

3

2Al  1.5O2 (0.0931 atm)

3000 2500 2Al  1.5O2  8.46Ar 2000 2Al  1.5O2  2.82Ar 1500 15

10

5

0

5

10

15

Assigned enthalpy (kJ/g) FIGURE 9.12 Adiabatic combustion temperature of a stoichiometric Al–O2 system containing various amounts of the inert diluent argon as a function of assigned enthalpy, other conditions being the same as those for Fig. 9.2.

510

Combustion

TABLE 9.3 Summary Data for Figure 9.12

Point

H°T (kJ/g)

T (K)

Ptot  PAr (atm)

Al2O3(l) (mole fraction)

Ar (mole fraction)

1

0.000

4005

1.0000

0.2160

0.0000

2

0.000

3769

0.3350

0.1178

0.5868

3

0.000

3527

0.0931

0.0736

0.8402

4

3.351

3527

0.0931

0.2271

0.7009

and aluminum nitride would obscure the major thermodynamic point to be made. As one can see from Fig. 9.12, the results with inert addition do not show a complete temperature plateau as the assigned enthalpy is varied. Next, a question arises as to whether the volatilization of the condensed-phase product oxide controls the combustion temperature. Analysis of points 1, 2, 3, and 4 specified on this figure verifies that it does. Table 9.3 lists the data for the explicit calculations at the four points. It is evident from the earlier discussion of the titanium–nitrogen system that the final volatilization temperature (3540 K) was controlled by the complete vaporization of the titanium formed due to the dissociation of the product titanium nitride. The partial pressure of titanium vapor was equal to 0.666 atm at the final volatilization temperature of 3540 K (see Fig. 9.8). Indeed, the vaporization temperature of titanium at 0.666 atm is 3530 K. Consequently, to analyze the results in Fig. 9.12, the conditions represented by points 3 and 4 were selected in order to consider what the gaseous partial pressures of the aluminum oxide vaporization components would be at the same temperature of 3500 K. Even though different amounts of assigned enthalpies were used in each case, the partial pressures of the oxide volatilization gases (the total pressure less the partial pressure of argon) are equal to 0.0931 atm in both cases. Figure 9.13 shows a plot of the calculated adiabatic flame temperature for the stoichiometric Al–O2 system as a function of pressure in the form used previously. The solid line labeled 2Al  1.5 O2 is based on the condition that no inert was added. What is significant is that points 3 and 4 on Fig. 9.13 were found to have a partial pressure of 0.0931 atm for the dissociated gases—the remaining gas contributing to the total pressure of 1 atm being argon. For a pressure of 0.0931 atm, Fig. 9.12 gives a temperature of 3500 K for a stoichiometric Al–O2 mixture. Similarly, for the non-argon gases, the partial pressure for point 2 is 0.3350 atm and the corresponding temperature is 3769 K. Point 2 falls directly on the ln P versus (1/T) line in Fig. 9.13. The significance of these correspondences is that the volatilization—or, more explicitly, the enthalpy of vaporization–dissociation determined from Fig. 9.2 of the condensed-phase

511

Combustion of Nonvolatile Fuels

10.0

Al

Pressure (atm)

Ti 1.0

1 1 2 2 2Al  1.5O2 (Al2O3)

0.1 3, 4 0.20

0.25

3 3Ti  2.5O2 (Ti3O5) 0.30

0.35

1000 (K)/T FIGURE 9.13 A plot similar to Fig. 9.11 to elucidate the significance of the points of Fig. 9.12.

5100

Temperature (K)

4200

3Ti  2.5O2

1 2

3300

3

3Ti  2.5O2  9.4N2

2400 3Ti  2.5O2  3.133N2 1500 10 5 0 5 Assigned enthalpy (kJ/g)

10

FIGURE 9.14 Adiabatic combustion temperature of a stoichiometric Ti–N2 system with various amounts of the diluent nitrogen as a function of assigned enthalpy (see Figs. 9.12 and 9.13).

A12O3 that forms—controls the flame temperature even when the gaseous mixture reacting with the metal is not pure oxygen. Figure 9.14 reports the data for titanium reacting with air and another O2–N2 mixture at 1 atm under stoichiometric conditions. The temperature–pressure variation for the pure O2–Ti system is also detailed on Fig. 9.13. Table 9.4 reports data similar to those in Table 9.3 for the aluminum system. Note that point 3 (Ti–air) has a combustion temperature of 3450 K and a partial pressure of the non-nitrogen decomposition gases of 0.10 atm. The Ti–O2 system

512

Combustion

TABLE 9.4 Summary Data for Figure 9.14

HoT (kJ/g)

T (K)

Ptot  PN2 (atm)

Ti3O5(l) (mole fraction)

N2 (mole fraction)

1’

0.0

4004

1.0000

0.0923

0.0000

2’

0.0

3768

0.4069

0.0771

0.5474

3’

0.0

3450

0.1000

0.0661

0.8453

Point

data in Fig. 9.13 reveal that at a pressure of 0.10 atm the adiabatic flame temperature should be 3471 K. For point 2, where the amount of nitrogen is onethird that in air, the combustion temperature is 3768 K and the partial pressure of non-nitrogen decomposition gases is 0.4069 atm. The temperature for this pressure on Fig. 9.13 is 3777 K. Thus, it is concluded that the formation of nitrogen oxides has a minimal effect on the controlling vaporization characteristics of the condensed-phase Ti3O5 that forms. When the N2 was replaced with the same molar concentration of Ar for the Ti–air case (3Ti  2.5O2  9.4N2), even though the specific heats of N2 and Ar are greatly different, the corresponding non-inert partial pressure was 0.1530 atm and the adiabatic flame temperature was 3558 K. The flame temperature calculated for the pure O2–Ti system at 0.1530 atm is also 3558 K, and the condition falls exactly on the P–T relationship for the Ti–O2 system in Fig. 9.13. These results for the metal reactions in air and oxygen–argon mixtures have great practical significance. In solid propellant rocket operation, the temperature of the flame around a burning aluminum particle will correspond to the volatilization temperature of the oxide at a pressure less than the total pressure; consequently, the radiative contribution to the propellant grain would not correspond to the known volatilization temperature at the total pressure, but to some lower temperature that would vary with the propellant composition. Furthermore, it is possible that some metals that would burn in the vapor phase in pure oxygen at 1 atm may burn heterogeneously in air. As mentioned in the previous section, the condition for vapor phase combustion versus heterogeneous combustion may be influenced by pressure by its effect on the flame temperature (Tvol or Td) as well as by its effect on the vaporization temperature of the metal reactant (Tb). For aluminum combustion in pure oxygen, combustion for all practical conditions occurs in the vapor phase. In air, this transition would be expected to occur near 200 atm as shown in Fig. 9.15 where for pressures greater than 200 atm, the vaporization temperature of pure aluminum exceeds the adiabatic flame temperature. This condition is only indicative of that which will occur in real particle combustion systems as some reactant vaporization will occur at temperatures below the boiling point

513

Combustion of Nonvolatile Fuels

6000 2Al  1.5O2

Temperature (K)

5000

2Al  1.5(O2  3.76N2) 2Al  3CO2,

Reactants at 298 K 4000

3000 2Al  3H2O(g), 2000

Al vaporization

0.1

1

2Al  3H2O(l),

Reactants at 398 K Reactants at 298 K

10 Pressure (atm)

100

FIGURE 9.15 Adiabatic flame temperatures and aluminum vaporization temperature as a function of pressure.

temperature of pure Al when inert and product species are present near the particle surface. In solid propellant rocket propulsion, aluminum is mostly oxidized by other oxygen containing oxidizers, such as CO2 and H2O, since for the generally fuel rich propellants, the amounts of O2 formed in the gaseous products is small (or rapidly consumed by the hydrocarbon binder). Oxidizers such as CO2 and H2O have also been considered for Mars lunar propulsion systems or for hydrogen production and underwater propulsion, respectively. Fig. 9.15 shows that when aluminum burns with either of these two reactants (with the reactants initially at near ambient conditions), the combustion process will burn heterogeneously at considerably lower pressures than with O2 due to the lower heats of reaction and consequently lower flame temperatures.

4. Combustion Synthesis The increasing importance of refractory materials has stimulated the search for new, economical techniques for synthesizing a number of substances that are not plentiful or pure enough to be useful. These materials are needed because new technologies require components capable of withstanding higher temperatures, more corrosive atmospheres, and increasingly abrasive environments. Material properties are important limiting factors in the design of cutting tools, turbine blades, engine nozzles, superconductors, semiconductors, electrodes, and certain components of nuclear power facilities. For those materials currently of practical importance, industrial production techniques require large amounts of external heat and involve complex, time-consuming processes. For example, the methods of forming the refractory compound titanium nitride

514

Combustion

(TiN) employ temperatures from 1400 to 1900 K and require anywhere from several hours to days. Additionally, most of these techniques require significant post-processing to achieve reasonable purity levels. A new development in the area of refractory material production began to receive considerable attention in the former Soviet Union in the mid-1970s. Led by Merzhanov and Borovenskaya [2] this work has been recognized as a major advancement and has created a new and important field of study in combustion involving mostly solid-phase (heterogeneous) reactions of metals. This new technique, designated self-propagating high-temperature synthesis (SHS), makes use of the exothermic nature of reactions that form ceramics and similar materials from their constituent elements. Using this process, carbides, borides, selenides, silicides, and sulfides have been fabricated by igniting a compacted mixture of their constituent powders (Table 9.5). Nitrides and hydrides have been produced by igniting powder compacts under nitrogen or hydrogen gas [12]. SHS offers many advantages in comparison with current commercial techniques. It requires much less external heat and far shorter reaction times; moreover, the purity of its products can be better than that of the initial reactants owing to the vaporization of impurities [13]. Initially, very poor yields were recovered using SHS when gaseous reactants N2 and H2 were involved. However, variations of the initial SHS process have improved yields. Such variations include replacing the gaseous nitrogen with a solid source of nitrogen (specifically, sodium azide) [13] and using high pressures in combination with dilution of the initial reactants with the product being produced [13, 14]. Indeed, a slurry of liquid nitrogen and titanium powder will, when ignited, propagate a thermal wave and create titanium nitride [15]. Unfortunately, the total mass of titanium necessary to cause thermal propagation of the wave is such that it is in excess stoichiometrically with the nitrogen; thus Ti cannot be completely converted to TiN. Many SHS processes include an oxide as one of the reactant materials. As a convenience, all the elements to be discussed in the various SHS processes will be referred to as metals. The selection of a metal–metal oxide reacting combination is readily made without detailed thermodynamic considerations. Nevertheless, it is fruitful to examine the overall thermodynamics that govern the choice of a particular SHS combination. Consider the classic thermite reaction Fe 2 O3  2 Al  Al2 O3  2 Fe  850 kJ

(9.4)

Intuitively, one knows that this equilibrium reaction will proceed to the right to form A12O3 and release heat. What largely determines the direction is the free energy change of the reacting system: ΔG   ΔH   T ΔS 

Group of metals of the periodic table Compounds Nitrides

II Mg3N2

III BN, AlN

IV

V

VI

VN, NbN, TaN(cub)

–

–

–

TiCx, ZrCx

TaN(hex), Ta2N, VC, NbC

WC

–

–

MnB

FeB NiB

–

–

–

–

–

–

HfC (x  0.6  1.0)

Nb2C, TaC

Borides

MgB2 MgB4 MgB6

–

TiB, TiB2 ZrB2, ZrB12 HfB2

VB, VB2, NbB, NbB2 TaB, TaB2

CrB, CrB2 MoB, Mo2B Mo2B5, W2B WB, WB2

Silicides

–

–

TiSi, TiSi2 ZrSi, ZrSi2

–

MoSi2

NbSe2

MoSe2

TaSe2

MoSe2

–

Solid solutions

TiC-WC, Ni  Al MoS2–NbS2 BC–BN NbC–NbN, TiC–TiN TaC–TaN, NbxZr1–xCyN1–y

Reported by Merzhanov and Borovenskaya [2].

TiSe2

WSe2

515

a

–

VIII

TiN, ZrN, HfN

Carbides

Chalco-genides

VII

Combustion of Nonvolatile Fuels

TABLE 9.5 Compounds Produced by Self-Propagating, High-Temperature Synthesis (SHS)a

516

Combustion

where the symbols are defined in Chapter 1. In most SHS processes the difference in the total number of moles of product formed compared to the total number of moles of reactant is small, particularly in comparison to that which occurs in the oxidation of a hydrocarbon fuel. Consequently, in SHS processes the order–disorder change is small and indicates that the entropy change must be small. In fact, the entropy change is quite small in these processes so that the TΔS term in the free energy equation above can be neglected even at the elevated temperatures that occur in SHS. Thus the free energy change is directly proportional to the enthalpy change and the enthalpy change becomes a measure of the direction an SHS reaction will proceed. Since an oxide product is always forming in an SHS system, it is evident that if a metal oxide with a smaller negative heat of formation is reacted with a metal whose oxide has a higher negative heat of formation, an exothermic reaction will occur, in which case the SHS process will proceed. If the opposite condition with respect to the heats of formation exists, the reaction will be endothermic and conversion will not occur. A good thermodynamic screening method for selecting an SHS process from among all the exothermic, and hence possible, metal–metal oxide systems is to consider the order of the heats of formation in terms of kilojoules per atom of oxygen as originally proposed by Venturini [16]. This procedure is feasible because ΔH° is determined from the heats of formation of the reactant and product oxides. Since the heats of formation of the elements are zero, the molar differences between the two oxides can be accounted for by dealing with the heats of formation per oxygen atom. An example of a molar difference would be 3TiO2  4Al → 2Al2 O3  3Ti

(9.5)

A list of oxides in these terms is presented as Table 9.6. Following the logic described, one obtains an exothermic system by choosing a metal whose oxide has a ΔH fo per oxygen atom high on the list to react with a metal oxide that is lower on the list. Indeed, the application of this conceptual approach appears to explain why in the oxidation of lithium–aluminum alloys, the dominant product is Li2O and that Al does not burn in the vapor phase [9]. The reaction sequence (9.4) must be ignited by an external heat source, even though the reaction is quite exothermic. The reason is that metals such as aluminum have a protective thin (35 Å) oxide coat. Until this coat is destroyed by another thermal source, the thermite reaction cannot proceed. Once the protective oxide is broken, reaction is initiated and the energy release from the initial reaction phase is sufficient to ignite the next layer of reaction, and so forth. The thermal wave established in these solid-phase reactions propagates in much the same way that a premixed gaseous flame propagates. Although some metals that react are not protected by oxide coats, ignition energy is necessary to initiate the reaction for the intermetallics.

517

Combustion of Nonvolatile Fuels

TABLE 9.6 Heats of Formation of Certain Oxides Oxide

ΔHfo at 298 K (kJ/mol)

Per oxygen atom

CaO

635

635

ThO2

1222

611

BeO

608

608

MgO

601

601

Li2O

599

599

SrO

592

592

Al2O3

1676

559

ZrO2

1097

549

BaO

548

548

UO2

1084

542

CeO2

1084

542

TiO2

945

473

SiO2

903

452

B2O3

1272

424

Cr2O3

1135

378

V2O5

1490

298

Fe2O3

826

275

WO3

789

263

CuO

156

156

For practical applications, one must calculate the temperature of the reaction in order to establish whether this temperature is so high that it will melt the metal that forms. Such melting causes metal flow which will prevent further reaction and hence propagation of the thermal wave. The actual temperature can be controlled by adding some of the product oxide to the reaction mixture. Consequently, for the system depicted as reaction (9.4), the initial reactants would be Fe2O3, Al, and A12O3. Sometimes, the production of very high temperatures works to advantage. If the temperature of an SHS is high enough to vaporize the metal formed, the reaction proceeds well. This type of reaction permits the titanium–sodium azide reaction to proceed to completion. The SHS technique found application in the system used to seed the van Allen belts during the 1968 International Geophysical Year. In this instance, the SHS

518

Combustion

system chosen was the reaction of barium with copper oxide with an excess of barium. Thus the products were barium oxide, copper, and barium. The temperature of the reaction was sufficient to ionize the barium in the product. Of course, Table 9.6 reveals that such a reaction will proceed. It is not necessary that one of the reactants be an oxide; it may, for example, be a chloride. Thus Table 9.7 represents a similar table to that for the oxides. The same logic for the choice of reacting systems prevails here [17]. Since the metal halides are readily vaporized, halogen exchange reactions can be used to liberate free metal atoms through gas-phase reactions, as opposed to the mostly heterogeneous “thermite-type” oxide reactions just discussed. In some early work by Olson et al. [18], a technique was developed for the production of solar-grade silicon based on this type of chemistry. These investigators examined the reaction between silicon tetrachloride and sodium vapor under conditions such that SiCl 4 (v)  4Na(v) → 4NaCl(v)  Si(l)

(9.6)

where (v) specifies vapor and (l) specifies liquid. The silicon liquid product comes from the nucleation of silicon atoms. Another SHS process is a modification of this halogen reaction to produce nitrides and other refractory and

TABLE 9.7 Heats of Formation of Certain Chlorides Chloride

ΔHfo at 298 K (kJ/mol)

Per chlorine atom

CsCl

443

443

KCl

437

437

BaCl2

859

430

RbCl

430

430

SrCl2

829

415

NaCl

411

411

LiCl

408

408

CaCl2

796

398

CeCl3

1088

362

AlCl3

706

235

TiCl4

815

204

SiCl4

663

166

519

Combustion of Nonvolatile Fuels

cermet powders. The simple addition of gaseous nitrogen to reactants similar to those above permits further reaction of the very reactive nascent metal atoms with the nitrogen to form the nitride. Table 9.8 gives the nitride information comparable to that in Table 9.7 for chlorides. From this information, one again observes that the metals that form useful metal nitrides have nitrides with large negative heats of formation and chlorides with relatively small negative heats of formation. Thermodynamically, because the heat of formation of nitrogen is zero, the titanium–nitrogen reaction must proceed, as has been found experimentally [19, 20]. Note that the opposite is true for the alkali metals, which have small nitride negative heats of formation and large chloride negative heats of formation. These comparisons suggest a unique method of forming nitrides via gas-phase reactions. For example, the reaction of an alkali-metal vapor, say sodium, with titanium tetrachloride, silicon tetrachloride, or aluminum tetrachloride vapor in the presence of nitrogen should produce titanium nitride, silicon nitride, or aluminum nitride according to the following overall reaction: MCl x  xNa  (y/ 2)N 2 → xNaCl  MN y

(9.7)

Mixed chlorides can also be used to produce intermetallics: MCl x  nM Cl y  (x  ny) Na → (x  ny) NaCl  MM n

(9.8)

where M represents various metal species.

TABLE 9.8 Heats for Formation of Certain Nitrides ΔHfo at 298 K (kJ/mol)

Per nitrogen atom

HfN

369

369

ZrN

365

365

TiN

338

338

AlN

318

318

BN

251

251

Mg3N2

461

231

Si3N4

745

186

Li3N

165

165

0

0

22

7

Nitride

N2 NaN3

520

Combustion

Thermodynamic calculations of the equilibrium product distribution from these alkali-vapor reactions reveal very poor yields with a large amount of sodium impurity owing to the very high flame temperatures involved and a correspondingly high degree of dissociation. Because of the very large latent heat of vaporization of sodium, the calculated results for liquid sodium were extremely promising, with conversion of metal to nitride in all three cases.

C. DIFFUSIONAL KINETICS In the case of heterogeneous surface burning of a particle, consideration must be given to the question of whether diffusion rates or surface kinetic reaction rates are controlling the overall burning rate of the material. In many cases, it cannot be assumed that the surface oxidation kinetic rate is fast compared to the rate of diffusion of oxygen to the surface. The surface temperature determines the rate of oxidation and this temperature is not always known a priori. Thus, in surface combustion the assumption that chemical kinetic rates are much faster than diffusion rates cannot be made. Consider, for example, a carbon surface burning in a concentration of oxygen in the free stream specified by ρmo. The burning is at a steady mass rate. Then the concentration of oxygen at the surface is some value mos. If the surface oxidation rate follows first-order kinetics, as Frank-Kamenetskii [21] assumed, Gox 

Gf  ks ρ mos i

(9.9)

where G is the flux in grams per second per square centimeter; ks the heterogeneous specific reaction rate constant for surface oxidation in units reflecting a volume to surface area ratio, that is, centimeters per second; and i the mass stoichiometric index. The problem is that mos is unknown. But one knows that the consumption rate of oxygen must be equal to the rate of diffusion of oxygen to the surface. Thus, if hDρ is designated as the overall convective mass transfer coefficient (conductance), one can write Gox  ks ρ mos  hD ρ (mo  mos)

(9.10)

What is sought is the mass burning rate in terms of mo. It follows that hD mos  hD mo  ks mos

(9.11)

ks mos  hD mos  hD mo

(9.12)

521

Combustion of Nonvolatile Fuels

⎡ h ⎤ D ⎥ mo mos  ⎢⎢ ⎥ k  h D ⎦ ⎣ s

(9.13)

⎡ ρk h ⎤ s D ⎥ Gox  ⎢⎢ ⎥ mo  ρ Kmo k  ⎣ s hD ⎦

(9.14)

or

K

ks hD ks  hD

k  hD 1 1 1  s   K khD hD ks

(9.15)

(9.16)

When the kinetic rates are large compared to the diffusion rates, K  hD; when the diffusion rates are large compared to the kinetic rates, K  ks. When k

hD, mos  mo from Eq. (9.13); thus Gox  ks ρ mo

(9.17)

When ks hD, Eq. (9.13) gives ⎛h ⎞ mos  ⎜⎜⎜ D ⎟⎟⎟ mo ⎜⎝ ks ⎟⎠

(9.18)

But since ks hD, it follows from Eq. (9.18) that mos

mo

(9.19)

This result permits one to write Eq. (9.10) as Gox  hD ρ(mo  mos )  hD ρ mo

(9.20)

Consider the case of rapid kinetics, ks hD, further. In terms of Eq. (9.14), or examining Eq. (9.20) in light of K, K  hD

(9.21)

522

Combustion

Of course, Eq. (9.20) also gives one the mass burning rate of the fuel Gf  Gox  hD ρ mo i

(9.22)

where hD is the convective mass transfer coefficient for an unspecified geometry. For a given geometry, hD would contain the appropriate boundary layer thickness, or it would have to be determined by independent measurements giving correlations that permit hD to be found from other parameters of the system. More interestingly, Eq. (9.22) should be compared to Eq. (6.179) in Chapter 6, which can be written as  H  ρ H mo  mo cp δ Lv cp ρ δ Lv α H D H  ρ mo  ρ mo δ Lv δ Lv

Gox 

(9.23)

Thus one notes that the development of Eq. (9.22) is for a small B number, in which case

hD 

D H δ Lv

(9.24)

where the symbols are as defined in Chapter 6. (H/Lv) is a simplified form of the B number. Nevertheless, the approach leading to Eq. (9.22) gives simple physical insight into surface oxidation phenomena where the kinetic and diffusion rates are competitive.

D. DIFFUSION-CONTROLLED BURNING RATE This situation, as discussed in the last section, closely resembles that of the droplet diffusion flame, in which the oxygen concentration approaches zero at the flame front. Now, however, the flame front is at the particle surface and there is no fuel volatility. Of course, the droplet flame discussed earlier had a specified spherical geometry and was in a quiescent atmosphere. Thus, hD must contain the transfer number term because the surface regresses and the carbon oxide formed will diffuse away from the surface. For the diffusioncontrolled case, however, one need not proceed through the conductance hD, as the system developed earlier is superior.

523

Combustion of Nonvolatile Fuels

Recall for the spherical symmetric case of particle burning in a quiescent atmosphere that ⎛ ρ D ⎞⎟ ⎟ ln(1  B) Gf  ⎜⎜⎜ ⎜⎝ rs ⎟⎟⎠

(9.25)

The most convenient B in liquid droplet burning was Boq 

imo H  c p (T  Ts )] Lv

(9.26)

since, even though Ts was not known directly, cp(T  Ts) could always be considered much less than imo H and hence could be ignored. Another form of B, however, is Bfo 

imo  mfs 1  mfs

(9.27)

Indeed, this form of B was required in order to determine Ts and mfs with the use of the Clausius–Clapeyron equation. This latter form is not frequently used because mfs is essentially an unknown in the problem; thus it cannot be ignored as the cp(T  Ts) term was in Eq. (9.26). It is, of course, readily determined and necessary in determining Gf. But observe the convenience in the current problem. Since there is no volatility of the fuel, mfs  0, so Eq. (9.27) becomes Bfo  imo

(9.28)

Thus, a very simple expression is obtained for surface burning with fast kinetics: ⎛ ρ D ⎞⎟ ⎟ ln(1  imo) Gf  ⎜⎜⎜ ⎜⎝ rs ⎟⎟⎠

(9.29)

Whereas in liquid droplet burning B was not explicitly known because Ts is an unknown, in the problem of heterogeneous burning with fast surface reaction kinetics, B takes the simple form of imo, which is known provided the mass stoichiometric coefficient i is known. For small values of imo, Eq. (9.29) becomes very similar in form to Eq. (9.22) where for the quiescent case hD  D/rs  α/rs.

524

Combustion

1. Burning of Metals in Nearly Pure Oxygen The concept of the B number develops from the fact that in the quasi-steady approach used for droplet burning rates, the bulk flow was due not only to the fuel volatilization but also to the formation of product gases. This flow, represented by the velocity v in the conservation equations, is outward-directed; that is, the flow is in the direction of increasing r. In the case of metal combustion in pure oxygen and at relatively high pressures, the possibility arises that heterogeneous processes may occur with no product gas volatilization. Thus the B number effect disappears and the bulk velocity is inward-directed. Indeed, it has been noted experimentally [22] that under these conditions small amounts of impurities in oxygen can reduce the burning rates of metals appreciably. The extent of this impurity effect is surprising and is worthy of examination. Consideration of Eq. (6.111) readily shows [23] for heterogeneous oxidation that there is no apparent gas-phase reaction. This equation is now written in the form ⎞ dYo d ⎛⎜  4πr 2 ρ vYo ⎟⎟⎟  0 ⎜⎜ 4πr 2 ρ D ⎟⎠ dr ⎝ dr

(9.30)

where the symbol Yo is now used for the mass fraction of oxygen to distinguish this unique case of droplet burning from all the others. Integrating Eq. (9.30) yields 4π r 2 ρ D

dYo  4πr 2 ρ vYo  c  m o dr

(9.31)

where the constant of integration is by definition the net mass flow rate of oxygen m o . Oxygen is the only species that has a net mass flow in this case, so m o  4πr 2 ρ v  4π(r 2 ρ v)s

(9.32)

where the subscript s, as before, designates the particle surface. The negative sign in Eq. (9.32) indicates that m o is inward-directed. Integrating Eq. (9.31) from r  rs to r   then yields ⎡ 1 Y ⎤ os ⎥ m o  ln ⎢⎢ ⎥ 1  Y o  ⎦ ⎣

(9.33)

where m o  m o /(4πρ Drs ) and (1  Yo) represents the initial impurity mass fraction. The metal surface reaction rate m f is now written as m f  4π r 2 ρsYos ks

(9.34)

525

Combustion of Nonvolatile Fuels

which is another representation of Eq. (9.17). Since mf  imo, it is possible to define from Eqs. (9.32) and (9.34) a nondimensional rate constant k such that s

4πr 2 ρs ks k s  i(4πρ Drs )

(9.35)

m o  k sYos

(9.36)

so that

One can see that k s is a form of a Damkohler number [ks/(Ds/rs)] which indicates the ratio of the kinetic rate to the diffusion rate. Substituting Yos from Eq. (9.36) into Eq. (9.33) yields the solution ⎡ 1  (m /k ) ⎤ o s ⎥ m o  ln ⎢⎢ ⎥ ⎢⎣ 1  Yo ⎥⎦

(9.37)

Knowing Yo and k s , one can iteratively determine m o from Eq. (9.37); and knowing m o one can determine the metal burning rate from m f  im o . The surface oxidizer concentration is given by Yos 

m o k

(9.38)

s

The next consideration is how small amounts of inert affect the burning rate. Thus, Eq. (9.37) is considered in the limit Yo → 1. In this limit, as can be noted from Eq. (9.37), m o k s . Thus rewriting Eq. (9.37) as ⎛ m ⎞⎟ e m o (1  Yo )  1  ⎜⎜⎜ o ⎟⎟  e ks (1  Yo ) ⎟ ⎜⎝ ks ⎠

(9.39)

and solving for m o , one obtains m o  k s ⎡⎢1  (1  Yo )e ks ⎤⎥ ⎣ ⎦

(9.40)

526

Combustion

Two observations can be made regarding Eq. (9.40). First, differentiating Eq. (9.40) with respect to the oxygen mass fraction, one obtains ∂m o = k s e ks ∂Yo

(9.41)

Thus one finds that m o varies in an exponentially sensitive manner with the ambient oxygen concentration, Yo, and consequently with the impurity level for a sufficiently fast surface reaction. Second, since k s is an exponential function of temperature through the Arrhenius factor, the sensitivity of the oxidation rate to the oxygen concentration, and hence the impurity concentration, depends on the metal surface condition temperature in an extremely sensitive, double exponentiation manner. The variation of the oxygen concentration at the surface can be clearly seen by rearranging Eq. (9.41) after recalling Eq. (9.38) to give ∂(m o /k s ) ∂Yos   ek ∂Yo ∂Yo

(9.42)

Figure 9.16 shows a plot of [(m /k )s  Yos ] versus Yo for different values of k s . The points on this figure were extracted from experimental data obtained by Benning et al. [22] for the burning of aluminum alloy rods in oxygen with an argon impurity. These data correspond to a k s close to 36. Large values of k s specify very fast surface reaction rates. For a k s value of 50, an impurity

1.0

m ˜ o/k˜s (Yos)

0.8

k˜s  0.5 1.0

0.6 5.0 0.4 0.2 50 0.0 0.94 0.95 0.96 0.97 0.98 0.99 1.00 Yo∞

FIGURE 9.16 A plot of the oxygen mass fraction Yos at the surface of a burning metal particle as a function of the ambient mass fraction Yo for the condition of Yo→1.

527

Combustion of Nonvolatile Fuels

mass fraction of 0.5% reduces the oxygen mass fraction at the surface to 0.1, a tenfold decrease from the ambient.

2. Burning of Small Particles – Diffusion versus Kinetic Limits Recall the diffusion controlled burning rate of a particle with fast heterogeneous reactions at the surface given by Eq. (9.29) Gf 

dr m f ρD  ρs s  ln(1  imo ) 4πrs2 dt rs

which may be integrated to obtain the combustion time from the initial particle size to burnout as t b,diff 

ρs d02 8ρ D ln(1  imo )

where d0 is the initial particle diameter, ρs the particle density, and ρD the product of the gas density and the diffusivity. This relationship is the “d2” relationship discussed in Chapter 6 for droplet combustion. As with droplet combustion, changes in oxidizer type are observed to affect the burning rate primarily by the diffusivity of the oxidizer and the overall reaction stoichiometry. For kinetically controlled combustion, Eqs. (9.9), (9.17), and (9.34) yield for the mass burning rate Gf 

dr m f  ρs s  ρs mo ks  iρ mo ks 4πrs2 dt

The combustion time from the initial particle size to burnout is t b,kin 

ρs d0 2i ρ mo ks

Thus, tb in a kinetically controlled regime is described by a “d1” law. Furthermore, tb is found to be inversely proportional to pressure (for a firstorder reaction) under kinetically controlled combustion, and in contrast, independent of pressure under diffusionally controlled combustion (since D  P1). In the kinetically controlled regime, the burning rate depends exponentially upon temperature.

528

Combustion

To determine the dominant combustion mechanism for a given set of conditions, the Damkohler number (Da) is given as

Da 

t b,diff t b,kin



d0 imo ks 4 D ln(1  imo )

If Da  1 is defined as the transition between diffusionally controlled and kinetically controlled regimes, an inverse relationship is observed between the particle diameter and the system pressure and temperature for a fixed Da. Thus, for a system to be kinetically controlled, combustion temperatures need to be low (or the particle size has to be very small, so that the diffusive time scales are short relative to the kinetic time scale). Often for small particle diameters, the particle loses so much heat, so rapidly, that extinction occurs. Thus, the particle temperature is nearly the same as the gas temperature and to maintain a steady-state burning rate in the kinetically controlled regime, the ambient temperatures need to be high enough to sustain reaction. The above equation also shows that large particles at high pressure likely experience diffusion-controlled combustion, and small particles at low pressures often lead to kinetically controlled combustion. Another length scale of importance to the combustion of small particles is the mean free path in the surrounding gas-phase. A comparison of this length scale to the particle diameter defines whether continuum conditions exist (i.e., the particle may be distinguished separately from the gas molecules). The key dimensionless group that defines the nature of the surrounding gas to the particle is the Knudsen number,

Kn 

2λ ds

where λ is the mean free path of the gas-phase. The mean free path for like molecules is given by

λ

1 2 πσ 2 N

where σ is the molecular diameter of the molecule and N is the gas concentration. Using kinetic theory to relate the gas viscosity to the mean free path yields,

529

Combustion of Nonvolatile Fuels

λ

2μ ⎛ 8 MW ⎞⎟1/ 2 P ⎜⎜ ⎜⎝ π RT ⎟⎟⎠

where μ is the gas viscosity and MW the molecular weight of the gas. The condition Kn  1 may also be used to roughly determine limiting regimes of particle combustion, which are effected by temperature and pressure through the mean free path. At atmospheric pressure, particles of dimensions 100 nm and smaller are characterized by Knudsen numbers greater than unity for the entire temperature range from room temperature to combustion flame temperatures, indicating that they can no longer be considered as macroscopic particles in a continuum gas. In the free molecular regime (Kn 1), nanopowders will in many ways behave similar to large molecules. Consequently, the reactivity of nanoparticles will generally be defined by kinetic rates and not transport rates of reactants (or energy) to the particle surface or products from the surface. As a result, considerable interest presently exists in the application of nanometer-sized metal particles to propellant combustion, where the Knudsen limit is attained for all temperatures. Much of the highly desirable traits of nano-sized metal powders in combustion systems have been attributed to their high specific surface area (high reactivity) [24, 25] and potential ability to store energy in surface defects [26]. In addition, nano-sized powders are known to display increased catalytic activity [27], superparamagnetic behavior [28], superplasticity [29], lower melting temperatures [30, 31], lower sintering temperatures [32], and higher theoretical densities compared to micron and larger sized materials. The lower melting temperatures can result in lower ignition temperatures of metals. For example, ignition temperatures of nAl particles have been observed to be as low as 1000 K (versus ignition temperatures closer to the melting temperature of alumina, common of micron-sized particles) [33, 34]. The combustion rates of materials with nanopowders have been observed to increase significantly over similar materials with micron-sized particles. For example, SHS reactions with nanopowders can support fast deflagrations and detonations (with combustion speeds of over 1000 m/s), which are several orders of magnitude greater than the propagation speeds of SHS reactions with micron and larger sized particles. A lower limit in the size of nano energetic metallic powders in some applications may result from the presence of their passivating oxide coating. For example, Al particles typically have an oxide coating with a limiting thickness of about 3 nm at room temperature. With a 100 nm diameter particle having a 3 nm thick coating, the energy loss per unit volume due to the presence of the oxide layer is 10%. A particle with a diameter of 10 nm would have an energy loss per unit volume of approximately 60%. Consequently, coatings, self-assembled monolayers (SAMs), and the development of composite

530

Combustion

materials that limit the volume of non-energetic material in the powders have been under development in recent years. An introduction to the combustion of nano-metallic particles may be found in a recent special issue of the Journal of Propulsion and Power [35].

3. The Burning of Boron Particles In certain respects, the combustion of boron is different from that of carbon because, under normal temperature and pressure conditions, the product oxide, B2O3, is not a gas. Thus, a boron particle normally has an oxide coat that thickens as the particle is heated in an oxidizing atmosphere. This condition is characteristic of most metals, even those that will burn in the vapor phase. For the efficient combustion of the boron particle, the oxide coat must be removed. The practical means for removing the coat is to undertake the oxidation at temperatures greater than the saturation temperature of the boron oxide B2O3. This temperature is about 2300 K at 1 atm. The temperature at which sufficient oxide is removed so that the oxidation can take place rapidly is referred to as the metal ignition temperature. The rate of oxidation when the oxide coat persists has been discussed extensively in Refs. [36, 37]. Nevertheless, what control the burning time of a boron particle is the heterogeneous oxidation of the clean particle after the oxide has been evaporated. Thus, for efficient burning the particle and the oxidizing medium temperatures must be close to the saturation temperature of the B2O3. Then the burning rate of the particle is given by Eq. (9.29), the same as that used for carbon except that the mass stoichiometric coefficient i is different. Even though the chemical reaction steps for boron are quite different from those of carbon, i is a thermodynamic quantity and the atomic weight 10 for boron is comparable to 12 for carbon; consequently, it is not surprising that the i values for both materials are nearly the same. Just as the surface oxidation chemistry makes it unlikely that carbon would yield CO2, it is also unlikely that boron would yield B2O3. Gaseous boron monoxide BO forms at the surface and this product is oxidized further to gaseous B2O3 by vapor-phase reactions. The gaseous B2O3 diffuses back to the clean boron surface and reacts to form three molecules of BO. The actual reaction order is most likely given by the sequence of reactions [37] discussed in the following paragraphs. In a high-temperature atmosphere created by the combustion of a host hydrocarbon fuel, there will be an abundance of hydroxyl radicals. Thus, boron monoxide reacts with hydroxyl radicals to form gaseous metaboric oxide HOBO. M  BO  OH → HOBO  M It is postulated that HOBO then reacts with BO to form gaseous boron oxide hydride, HBO, and boron dioxide, BO2 (OBO).

531

Combustion of Nonvolatile Fuels

OBOH  BO → OBO  HBO The boron dioxide then reacts with another BO to form boron oxide B2O3. OBO  BO → B2 O3 This route is consistent with the structure of the various boron oxide compounds in the system [37]. In a hydrogen-free oxidizing atmosphere, a slower step forms the boron dioxide, BO  O2 → OBO  O whereupon B2O3 again forms via the reaction above. After the gaseous reaction system is established, the B2O3 diffuses back to the nascent boron surface to form BO, just as CO2 diffuses back to the carbon surface to form CO. The reaction is B  B2 O3 → 3BO Thus, the overall thermodynamic steps required to calculate the mass stoichiometric index in Eq. (9.29) are 2 BO  21 O2 → B2 O3 B  B2 O3 → 3BO B  21 O2 → BO and i  10/16  0.625 compared to the value of 0.75 obtained for the carbon system. Boron does not meet Glassman’s criterion for vapor-phase combustion of the metal. Thus, the boron surface remains coated with a vitreous B2O3 layer and boron consumption becomes extremely slow; consequently, boron is not burned efficiently in propulsion devices.

4. Carbon Particle Combustion (C. R. Shaddix) The appropriate stoichiometric coefficient for oxidation of carbon is not readily apparent, because there are two different oxidation states of carbon, namely carbon monoxide and carbon dioxide, which may be present when a carbon surface is oxidized. These products form according to the overall reaction steps C(s)  O2 → CO2

(ΔH  394 kJ/mol C)

532

Combustion

and 2C(s)  O2 → 2CO

(ΔH  110 kJ/mol C)

The large differences in the heat release of these reactions and the twofold difference in the carbon gasification rate per reacted O2 molecule make it imperative to properly understand the oxidation reaction. Consequently, a number of experiments have been performed to attempt to shed light on this reaction path. Complicating the interpretation of these experiments is the fact that CO readily oxidizes to CO2 at high temperatures (particularly if there is any moisture present, as discussed in Chapter 3). The dominant viewpoint arising from these studies is that the direct production of CO2 from carbon oxidation only is significant at fairly low temperatures, such that it is usually safe to assume under combustion conditions that only CO is produced. With this assumption, the value of i in Eq. (9.28), that is, the mass stoichiometric coefficient for fuel to oxygen, is equal to 24/32 or 0.75. In the simplest situation, the CO produced at the particle surface diffuses away from the surface without further reactions. This assumption, known as the single-film or frozen boundary layer model, results in the species profiles shown in Fig. 9.17. For sufficiently high temperatures (greater than 1000 K) and sufficiently large particles, the CO will oxidize in the particle boundary layer according to the overall reaction 2CO  O2 → 2CO2

T O2

YO2

CO YCO

.

r ∞

r  rp r

FIGURE 9.17 Schematic of the “single-film” model of carbon particle combustion, whereby oxygen and carbon monoxide counterdiffuse through an unreactive boundary layer.

533

Combustion of Nonvolatile Fuels

When this happens, the oxygen flux to the particle surface is cut-off because of its consumption in oxidizing CO, and CO2 becomes the relevant gasification agent for the particle surface according to the Boudouard Reaction: CO2  C(s) → 2CO

(ΔH  172 kJ/mol C)

When these conditions exist the carbon is consumed according to a doublefilm model with species profiles as indicated in Fig. 9.18. Carbon dioxide oxidizes carbon at a substantially slower rate than O2 at normal combustion temperatures. As a consequence, the transition from single-film combustion of a carbon particle to double-film combustion typically involves a strong reduction in the carbon oxidation rate, as eloquently demonstrated by Makino and coworkers in a series of experiments in which graphite rods were oxidized in air at different temperatures and flow rates [38]. For oxidation of solid carbon to produce CO (either by O2 or CO2), two moles of gaseous combustion products are produced per mole of gaseous reactant, resulting in a net gas flow away from the particle surface. This Stefan flow reduces the rates of heat and, especially, mass transfer from the boundary layer to the particle surface (akin to drag reduction on a flat surface with

CO2

O2

T T CO2

YO2

CO YCO

.

YCO2

YCO2 r∞

r  rp r

FIGURE 9.18 Schematic of the “double-film” model of carbon particle combustion, whereby carbon monoxide produced at the particle surface is oxidized to carbon dioxide in a boundary layer flame, consuming the oxygen that is diffusing toward the particle.

534

Combustion

active blowing). Consequently, accurate modeling of carbon oxidation must account for Stefan flow reduction of the diffusion rate of oxygen to the particle surface.

E. PRACTICAL CARBONACEOUS FUELS (C. R. SHADDIX) Solid carbonaceous fuels such as coal, biomass, and petroleum coke are widely used throughout the world to provide heat and to generate electrical power through combustion processes. These fuels are generally classified according to their heating value (energy content), volatile fuel content, fixed carbon content, moisture level, and ash or mineral content. The volatile fuel content is the fraction of the original fuel mass that evolves as gases when the fuel is heated to high temperatures, whereas the fixed carbon is the mass of carbon that remains after the volatiles have escaped. The ash content refers to the portion of the solid that remains when the fixed carbon has been fully oxidized, leaving behind oxidized mineral compounds. For some coal and biomass sources, the ash content can be very large and pose a severe hindrance to the effective combustion of the fuel. Furthermore, the formation of ash deposits, particularly those that melt (referred to as “slagging” deposits) or otherwise are difficult to remove, restricts the effective heat transfer from furnace gases to boiler tubes and plays an important role in boiler design and operation. Excessive moisture in the fuel reduces the fuel heating value and flame stability. Conversely, insufficient fuel moisture can lead to spontaneous ignition problems when storing and handling some reactive fuels.

1. Devolatilization The devolatilization process (referring to the release of gaseous fuel components as the solid fuel is heated) is a key characteristic of the combustion of solid fuels. The volatile gases burn much more rapidly than the remaining char particles and therefore are important for flame ignition and stability and play an important role in the formation of oxides of nitrogen (NOx), a regulated class of pollutants. Moreover, the devolatilization process determines how much char remains to be burned as well as the physical characteristics of the resulting char, with subsequent impacts on the char combustion properties. Different coal types vary significantly in their volatile content, ranging from a maximum of approximately 50% (by mass) for low- and mid-rank coals to just a few percent for anthracitic coals (which are graphite-like in character). Biomass always has a large volatile content, generally around 80%, as determined using the standard ASTM test method. Figure 9.19 shows a characteristic molecular structure of coal, featuring an aromatic carbon backbone and a wide range of bond strengths. General plant matter (as distinct from the fruiting bodies that are often used as food and are

535

Combustion of Nonvolatile Fuels

H2

H C H

H2 H

H2

H H H2 OH

H2

O

H C H O

OH H C CH3

H C H CH3

H C H O

H

H2 O OH H2 H CH

N

H C H

H2

C OH C

S

H

H O CH3 H C

H2

O

O

H2

CH3

CH3

O

H2

H H2

SH

H

H

CH3

H2 H

H2

H OH C H

H2

H2 H C H O

O

H2

H C H

H2

N H2

FIGURE 9.19 Hypothetical coal molecule as represented by Solomon [39]. Reprinted with permission from the American Chemical Society.

CH2OH H

H

O

O

H OH H

OH H H H O

H H

CH2OH

OH H

H O

O

H OH H H

CH2OH

OH

H

O

H

OH

OH H H H O

H O

CH2OH

OH

(a)

OH

OH

O H HO

OH

H

OH

O H

O HO

OH

H

OH

O H

O HO n

OH

H O HO

O H

H O

OH

(b) FIGURE 9.20 Chemical structure of cellulose (a), which is a glucose polymer, and xylan (b), a typical component of hemicellulose.

primarily composed of starch and sugar) is known as lignocellulosic biomass and is composed of a more or less even mixture of cellulose, hemicellulose, and lignin. As shown in Fig. 9.20, cellulose and hemicellulose are both composed of polymers of oxygen-containing ring compounds linked by relatively

536

Combustion

weak carbon–oxygen bonds. Lignin, in contrast, is composed of small aromatic units connected in weakly linked branched structures. As coal and biomass particles are heated, the internal structure of the carbonaceous material undergoes internal molecular rearrangements. Many weakly bonded moieties break their connecting bonds to the main structure and form gas molecules that aggregate within the solid and, after building sufficient pressure, burst forth from the particle with substantial force. At the same time, some weakly bonded structures and some structures with intermediate-strength bonds pivot about their connecting bond and are able to form stronger, cross-linking bonds with neighboring regions of the structure. Thus there is an inherent competition between solid decomposition and char-forming reactions, with different characteristic activation energies and reaction times. As a consequence, the quantity and chemical composition of the volatile matter that is emitted from these fuels is highly dependent on the nature of the original fuel structure, the rate at which the particles are heated, and the final temperature attained by the particles. For large particles, this also means that the devolatilization process differs as a function of the internal radius of the particle, because the local heating rate varies with the distance from the surface of the particle, where heat is being applied. The effect of heating rate on evolution of volatiles is most clearly evidenced in the case of woody biomass, which has been shown to have a volatile yield of greater than 90% when small particles are rapidly heated to 1200°C and to have a volatile yield of only 65% when large particles are slowly heated to 500°C in the commercial charcoal-making process. With the importance of the devolatilization process to solid particle combustion and the complexity of the chemical and physical processes involved in devolatilization, a wide variety of models have been developed to describe this process. The simplest models use a single or multiple Arrhenius rates to describe the rate of evolution of volatiles from coal. The single Arrhenius rate model assumes that the devolatilization rate is first-order with respect to the volatile matter remaining in the char [40]: dV  k (V  V ) dt k  BeE/RT

(9.43)

Fits of Eq. (9.43) to the experimental data typically yield an effective activation energy of about 230 kJ/mol, which is consistent with the activation energy for rupturing an ethylene bridge between aromatic rings [41]. The single-rate approach defined by Eq. (9.43) adequately captures the increasing rate of devolatilization at higher temperatures, but fails to account for the observed change in the volatiles yield as a function of temperature. To capture this, Kobayashi et al. [42] proposed the use of two competing reaction

537

Combustion of Nonvolatile Fuels

paths with different activation energies and different volatile yields (with the higher activation energy path having a higher volatile yield). This “Kobayashi Model” is expressed schematically as k1

V1  (1V1)char

k2

V2  (1V2)char

Coal

with k1  B1eE1/RT, k2  B2eE2/RT, and V1 and V2 referring to the volatiles yield along each reaction path. With this model, the instantaneous devolatilization rate is the sum of the two independent rates, as shown in Eq. (9.44). dc  (k1  k2 )c dt dV  (V1k1  V2 k2)c dt

(9.44)

where c refers to the mass of solid coal or char remaining. Another approach is known as the Distributed Activation Energy Model (DAEM). This model recognizes that devolatilization occurs through many simultaneous reactions. To express this process in a mathematically tractable manner, these reactions are all presumed to be first order and to be describable by a continuous distribution of kinetic rates with a common pre-exponential and a defined distribution function of activation energy [43].  ⎛ t ⎞⎟ V  V ⎜  ∫ exp ⎜⎜∫ k (E )dt ⎟⎟⎟ f (E )dE ⎜⎜⎝ ⎟⎠ V 0 0

(9.45)

If a Gaussian distribution is used for the activation energy, the distribution function has the form f (E ) 

exp ⎡⎣⎢(E  E0 )2 / 2σ 2 ⎤⎦⎥ σ (2π )1/2

(9.46)

and the distribution of kinetic rates is characterized by a mean and standard deviation of the activation energy, so only one additional parameter has been introduced relative to the simplistic approach of using a single Arrhenius rate [Eq. (9.43)]. Advances in the understanding of coal structure and its evolution during devolatilization have led to the development of several coal network models

538

Combustion

that give predictions of volatile mass evolved as well as predictions of the chemical speciation of the released volatiles [44–46]. Detailed speciation of volatiles is important for understanding NOx and soot formation as well as flame ignition and stability. Extensions of these models to cover biomass fuels have recently been developed, although in some cases the dispersed alkali metal content (primarily potassium) in biomass plays a significant role in catalyzing the reaction steps during devolatilization, making accurate predictions difficult. The earliest network model came to be known as the functional group depolymerization, vaporization, and cross-linking (FG-DVC) model [44]. The FG portion of the model considers that certain functional groups in the coal lead to the formation of light gas species upon heating. The DVC portion of the model describes the deconstruction of the original macromolecular network through bridge-breaking reactions that yield tar and cross-linking reactions that produce char. The FG-DVC model relies heavily on data from thermogravimetric analysis (mass loss during heating) coupled with Fourier transform infrared (FTIR) spectrometer analysis of the evolved gases and uses Monte Carlo simulation of lattice statistics. A significantly different approach to modeling coal devolatilization is known as chemical percolation devolatilization (CPD) [45]. This approach relies on a detailed description of the chemical structure of the coal, as provided by solid-state nuclear magnetic resonance (NMR) spectroscopy data, which characterizes the fraction of carbon atoms with different types of bonds. The NMR data can be used to infer the average molecular weight of fused aromatic clusters and the number of branching sites for the cluster. In the CPD approach, the NMR data are coupled with percolation statistics to describe the decomposition of the original array of aromatic ring clusters into side chains and reactive intermediates. The side chains convert to light gases and the reactive intermediates undergo a competitive reaction scheme to either form tar or generate light gases and char. The last of the three major network models is known as FLASHCHAIN [46]. This approach uses a simplified description of the original coal structure as a linear chain, which decomposes according to chain statistics into a broad distribution of fragment sizes. The fragments are partitioned into volatile or condensed species according to mixture phase equilibrium (i.e., flash distillation). Char formation occurs in the condensed phase with concurrent evolution of fixed gases. Input parameters in the original formulation of this approach included the coal ultimate analysis (i.e., atomic composition), NMR data, and extract yield, but later studies demonstrated that good predictability is still achieved by only using the ultimate analysis and basing the values of the other input parameters on correlation functions. The ambient pressure has a complex effect on the devolatilization process. For one thing, in most practical situations an increase in ambient pressure will tend to increase the coal particle heating rate for a given reactor temperature.

539

Combustion of Nonvolatile Fuels

This effect, in and of itself, tends to increase the yield of tar (and therefore of total volatiles), for the reason discussed earlier. However, increasing the ambient pressure also shifts the vapor–liquid equilibrium of the tar species to smaller tar species (with higher vapor pressures) and thus tends to diminish the overall release of tar. Wire-mesh experiments with well-controlled particle heating rates show a significant reduction in the yield of tar and total volatiles as the pressure is increased. The rate of devolatilization, however, is nearly insensitive to pressure, as would be expected for unimolecular reaction processes.

2. Char Combustion Once devolatilization of the solid fuel has completed, a porous char particle remains and is consumed through surface reactions of oxidizing species such as O2, H2O, and CO2, as discussed earlier in Section D. The presence of pores in the char particle allows for penetration of reactant species into the particle and therefore for much greater surface area for reactions than is associated with the external surface of the particle. Char particles often have porosities or void fractions of greater than 0.3, depending on the amount of volatiles in the original particle and the extent of particle swelling or shrinkage during devolatilization. Internal surface areas of char particles often exceed 100 m2/g. Using a typical apparent char density of 0.8 g/cm2, one can show that the internal surface area exceeds the external surface area of a particle by over a factor of 10 for a 1 μm particle, by over a factor of 1000 for a 100 μm particle, and by over a factor of 10,000 for a 1 mm particle (the ratio of surface areas scales by particle diameter). The extent to which a given reactant, such as oxygen, is able to utilize this additional surface area depends on the difficulty in diffusing through the particle to reach the pore surfaces and on the overall balance between diffusion control of the burning rate and kinetic control. To broadly characterize these competing effects, three zones of combustion of porous particles have been identified, as shown in Fig. 9.21. In Zone I the combustion rate is fully controlled by the surface reaction rate (kinetically controlled), because the diffusion

Zone I

Zone II

YO

YO

2, ∞

YO

2, ∞

YO

2

YO

2

rp r

∞

0

2

YO

2

0

YO

2

YO

0

Zone III

YO

2, ∞

2

0

rp r

∞

0

0

rp

∞

r

FIGURE 9.21 Sample oxygen profiles through the char particle boundary layer and the char particle itself during combustion proceeding according to the characteristic burning zones.

540

Combustion

rate of oxygen is so fast (relatively) that the oxygen concentration throughout the particle and through the particle’s boundary layer is essentially equal to the bulk gas oxygen content. Zone I combustion is favored by low temperatures (reducing the surface reaction rate) and small particle sizes (enhancing the surface-specific diffusive flux to the particle). In Zone II the combustion rate is determined by the combined effects of the surface reaction rate and the diffusion of oxygen (both to the particle and through the interior of the particle). The large internal surface area of most char particles coupled with a wide range of pore sizes and different degrees of access to the pores means that this zone is active over a wide range of combustion conditions. In Zone III, the surface-burning rate is so fast that oxygen does not effectively penetrate the particle before being consumed. This is diffusion-limited combustion, as was described in Section D3. Zone III combustion is favored by high temperatures (increasing the surface burning rate) and large particles (reducing the surfacespecific diffusive flux). In one sense, Zone I and Zone III modes of combustion are simply limiting extremes of the general condition of Zone II combustion. Zone I combustion proceeds at an overall rate equal to the product of the intrinsic burning rate, evaluated at the ambient oxygen concentration, and the total internal surface area. The char diameter necessarily stays constant and the particle density continually decreases as particle mass is evenly removed throughout the particle on the pore surfaces (constant-diameter combustion). In Zone III combustion, the burning rate is determined by the diffusive flux of oxygen through the particle boundary layer. The particle density remains constant throughout burnout and the particle size continually decreases as mass is removed solely from the particle surface (constant-density combustion). Zone II combustion proceeds with partial penetration of oxygen, resulting in simultaneous variations in particle density and diameter as the pores closest to the particle surface undergo oxidation, in addition to the external surface of the particle. The ratio of the actual burning rate to the maximum possible rate if the entire particle was subject to the oxygen partial pressure at the external particle surface is known as the effectiveness factor.

3. Pulverized Coal Char Oxidation The predominant way in which coal is used to generate steam and electrical power is by means of pulverizing the coal to a fine dust that readily ignites and burn when introduced into a high-temperature furnace. The small particle sizes mean that the coal is readily entrained into the furnace flow and is able to complete combustion before exiting the furnace. Pulverizing the coal also allows a significant portion of its combustion to be completed in the near-burner region, facilitating the use of air-staging and reburning techniques to reduce NOx production. Another significant advantage of pulverized coal combustion relative to other coal combustion techniques, such as fluidized bed combustion, is the ease of scaling boiler sizes to produce upwards of 600 MW of electrical power

541

Combustion of Nonvolatile Fuels

from a single boiler, thereby improving the boiler efficiency and reducing capital costs. Modern pulverized coal boilers can produce electrical power with an overall thermal efficiency of approximately 45%, through the application of a supercritical steam cycle (requiring the use of high-temperature steel alloys in the boiler superheater). Improvements in the manufacture of cost-effective high-temperature steel alloys may improve the thermal efficiency to greater than 50%. A standard commercial pulverized coal grind usually is required to have 70 wt.% of the particles with a size less than 200 mesh (74 μm) and less than 2% over a 50-mesh size (300 μm). With these small particle sizes and the high temperatures within pulverized coal boilers, the coal char particles generally burn in Zone II with combined control of diffusion and chemical reaction. Traditionally the combustion kinetics of pulverized coal have been described through a global particle kinetics approach which calculates the particle reaction rate based on the external surface area of the particle and the concentration of oxygen at the particle surface (thereby accounting for the diffusion profile of oxygen through the particle boundary layer). The dependence of the reaction rate on the partial pressure of oxygen is usually expressed as a power law, yielding a rate expression that is referred to as an “nth-order, Arrhenius” apparent kinetic expression: w  AeE/RTp POn

2 ,s

(9.47)

where w is equal to the instantaneous burning rate of the particle divided by its external surface area (  πd p2 ) and is usually expressed in units of kg/(m2 s). According to an analysis of reactions in idealized porous catalysts first reported by Thiele [47] in 1939, in Zone II combustion the apparent activation energy, E, and the apparent reaction order, n, appearing in Eq. (9.47) may be related to the actual, intrinsic kinetic parameters as E  Eint/2 and n  (m  1)/2 where Eint is the intrinsic activation energy and m is the intrinsic reaction order. The intrinsic activation energy for carbon or char oxidation has been determined to be approximately 180 kJ/mol and the intrinsic reaction order has been measured (at low to intermediate temperatures) to lie between 0.6 and 1.0 [48]. Smith [49] correlated the oxidation data for a range of carbon types, including porous chars and impervious carbon such as soot to derive the following expression for the intrinsic char oxidation kinetics for an oxygen partial pressure of 101 kPa: ⎛ 179.4 ⎞⎟ w  3050 exp ⎜⎜ ⎟ ⎜⎝ RT ⎟⎠

(9.48)

Because of the variability in deduced reaction orders for different experiments and carbon types, a general expression for the kinetic rate that includes the oxygen dependence could not be determined.

542

Combustion

In reality, it is believed that the oxidation of carbonaceous surfaces occurs through adsorption of oxygen, either immediately releasing a carbon monoxide or carbon dioxide molecule or forming a stable surface oxygen complex that may later desorb as CO or CO2. Various multi-step reaction schemes have been formulated to describe this process, but the experimental and theoretical information available to-date has been insufficient to specify any surface oxidation mechanism and associated set of rate parameters with any degree of confidence. As an example, Mitchell [50] has proposed the following surface reaction mechanism: 2Cf  O2 → C(O)  CO 2Cf  O2 → C2 (O2 ) Cb  Cf  C(O)  O2 → CO2 + C(O) + Cf Cb  Cf  C(O)  O2 → CO  2C(O) Cb  C(O) → CO  Cf Cb  C2 (O2 ) → CO2  2Cf where an O or O2 in parentheses denotes a surface oxide, Cf represents a surface carbon atom with an open bond, and Cb represents a “bulk” carbon atom (i.e., one with fully assigned bonds).

4. Gasification and Oxy-Combustion Fuel-rich oxidation of solid carbonaceous fuels to form gaseous fuel components such as methane, hydrogen, and carbon monoxide is known as gasification. Coal gasification has been an industrially important technology for many decades and continues to be used for the generation of synthetic natural gas, hydrogen, ammonia, and specialty chemicals. Recently, gasification of coal has arisen as an important technology for high efficiency generation of electrical power using large aeroderivative gas turbine engines with a bottoming steam power cycle. This technology, known as integrated gasification combined cycle (IGCC), can currently achieve overall thermal efficiencies near 45% with very low emissions of traditional pollutants such as NOx, SO2, and mercury. In addition, by converting the carbon monoxide in the gasifier product gas to carbon dioxide and hydrogen according to the water–gas shift reaction, CO  H 2 O → CO2  H 2

(ΔH  41 kJ/mol)

gasification of coal and biomass can yield large quantities of relatively inexpensive hydrogen and readily allow for capture and subsequent sequestration of the fuel carbon (in the form of CO2). A typical system layout for coal or biomass gasification to produce hydrogen and electricity while capturing carbon is shown in Fig. 9.22. Co-gasification of coal and biomass with

543

Combustion of Nonvolatile Fuels

N2 Air

H2 gas turbine

H2-rich syngas

Air separation N2 unit

air

O2 Syngas

Coal Biomass Gasifier

Gas clean-up

Shift reactor

CO2 separation

H2 separation

Steam turbine

Steam CO2

Slag

Sequestration or use

H2 Storage/ fuel cell/ gas turbine

FIGURE 9.22 Layout for a gasification-based system for co-production of hydrogen and electrical power while sequestering CO2 [51]. Reprinted with permission from Elsevier.

carbon capture and sequestration is a promising technology for power and/or fuel production that is not merely carbon-neutral but can be carbon-negative. Gasification of solid fuels also allows for conversion of the solid fuel feedstock into one or more liquid fuels by processing the gasifier syngas (cleaned of impurities such as H2S, HCN, and NH3) over catalytic reactors to yield methanol, ethanol, or even gasoline and diesel fuel, through the Fischer–Tropsch catalytic synthesis process originally developed in Germany. Analogous to solid fuel boilers, gasifiers can be operated either as fluidized bed devices with relatively large fuel particles, moderate temperatures, and long particle residence times, or as entrained flow devices with pulverized fuel particles, high temperatures, and short residence times. Either air or oxygen can be used as the primary oxidant gas, although steam is generally added to assist in char gasification, according to the following reaction: H 2 O  C(s) → CO  H 2

(ΔH  130 kJ/mol C)

In addition to char gasification from steam, CO2 produced locally from consumption of oxygen in the gasifier reacts with char according to the Boudouard Reaction (previously shown). The generation of CO and CO2 from the oxygen consumed in the gasifier releases the energy required to devolatilize the raw fuel particles and gasify the resultant char through the endothermic reactions with steam and CO2. Air-blown gasifiers benefit from a low-cost (free) source of oxidant, but produce a gasifier product gas that has a very low heating value (approximately 5 MJ/m3 for air-blown gasification of coal, compared to 38 MJ/m3 for a typical natural gas) and which is too dilute to use for liquid fuel synthesis.

544

Combustion

Oxygen-blown gasifiers yield a moderate heating value product gas (approximately 9 MJ/m3) and can operate at very high temperatures, but suffer from the significant cost of producing oxygen with an air-separation unit. Because CO2 and H2O react much slower with char than O2, gasifiers usually operate at elevated pressure (to increase the collision rate of the gasifying species with the char). For IGCC systems, operating the gasifier at elevated pressure eliminates the need for additional compression before injection of the product gases into the gas turbine combustor. Biomass gasifiers are typically operated at atmospheric pressure because of the difficulty of feeding biomass particles into a pressurized gasifier and because the high biomass volatile content and high reactivity of biomass chars obviate some of the need for elevated pressures to obtain good carbon conversion. An alternative technology to gasification for power production with carbon capture and sequestration is to burn the solid fuel in oxygen, instead of air. To moderate the combustion temperatures and avoid the formation of molten, slagging deposits of mineral matter on the boiler tubes, the oxygen is usually diluted with recycled flue gas before entering the combustion chamber. The very high concentration of carbon dioxide in the exhaust gas from this process allows fairly simple capture of CO2 (the major additional financial cost and efficiency loss in this system results from the oxygen generation process). In addition, compression of the flue gas for CO2 transport allows capture of pollutant species, making this a near zero-emission process. In contrast to the option of installing new gasification power plants, the oxy-fuel combustion technology is generally believed to be retrofitable to existing boiler systems, thereby taking advantage of the large capital investment that has already been made in these boilers. A critical consideration in oxy-fuel combustion systems is the extent of flue gas recirculation that is desired. From a cost perspective, one would like to minimize the amount of recirculation, thereby increasing the oxygen concentration of the gases entering the boiler. However, the volatile flame temperature and char combustion temperature both increase substantially with higher oxygen content. Counteracting this influence of increasing oxygen content is the substitution of CO2 for N2 as the predominant gas in the boiler. Carbon dioxide has a molar specific heat that is 1.7 times larger than that of molecular nitrogen. Therefore, combustion in CO2 environments results in lower flame temperatures than combustion in equivalent N2 environments. Figure 9.23 shows the computed adiabatic flame temperature for stoichiometric combustion of methane in mixtures of oxygen with nitrogen and oxygen with carbon dioxide. As is evident in this figure, combustion of methane in 32% O2 in CO2 produces the same flame temperature as combustion in dry air (with 21% O2 in N2). A further, important consideration for boiler operation is the thermal radiation emitted by hot CO2 in the furnace – in contrast to N2, CO2 emits thermal radiation in the infrared and can impact the balance between radiant and convective heat transfer in a boiler. In addition, the flue gas recycle substantially increases the water vapor content of the furnace gases, thereby enhancing thermal radiation from hot H2O.

545

Combustion of Nonvolatile Fuels

2600

N2

Tad (K)

2400 Tad in air 2200 2000

CO2

1800 1600

20

25

30

35

40

Vol-% O2 FIGURE 9.23 Adiabatic flame temperature for stoichiometric combustion of methane in mixtures of oxygen with nitrogen and oxygen with carbon dioxide, computed using NASA’s Chemical Equilibrium Analysis (CEA) program [52].

For application to combustion of solid fuels, the presence of high concentrations of CO2 decreases the volatile flame temperature (for a given O2 level) because of the higher specific heat of CO2. Also, CO2 decreases the volatile combustion rate because of the reduced diffusivity of O2 through CO2 atmospheres (the diffusivity is about 20% lower). For char combustion, high levels of CO2 decrease the burning rate during Zone II and Zone III combustion because of the lower oxygen diffusivity. It is possible that CO2 enhances the overall kinetic rate of carbon oxidation through contributions from the Boudouard Reaction, but it is generally believed that this reaction is too slow to effectively compete with char oxidation from O2, even at the very high particle temperatures that are sometimes experienced with the use of high oxygen concentrations. NOx formation, both from oxidation of nitrogen in the volatile fuel components and oxidation of the nitrogen in the char, is enhanced for the high volatile flame temperatures and elevated char combustion temperatures associated with enhanced oxygen levels. However, practical oxy-combustion systems recycle a substantial fraction (typically on the order of 70%) of the flue gas back to the combustion furnace, allowing significant reburning of the previously formed NOx in the flame zone and on the burning char particles. Also, substitution of CO2 for N2 eliminates the contribution of thermal NOx formation. As a result, the net NOx production is reduced for oxy-fuel combustion processes, typically by a factor of three or more.

F. SOOT OXIDATION (C. R. SHADDIX) A detailed description of the chemical processes involved in soot formation was given in the previous chapter. Oxidation of soot particles is an important subject area because in most practical processes that form soot (e.g., diesel

546

Combustion

engines, gas turbines, and boilers) much more soot is formed in the flames than is emitted as a pollutant. In other words, it is the incomplete oxidation of soot within flames that leads to soot emissions. Also, the oxidation rate of soot controls the amount of soot that exists in the highest temperature regions of a flame and thereby has a large influence on the radiant energy emitted from flames. Most boiler and furnace systems (especially glass melting furnaces) rely on thermal radiation from hot, oxidizing soot to transfer heat effectively to the walls or other heat-absorbing surface. As carbonaceous particles composed primarily of carbon, the chemical oxidation of soot proceeds in a similar manner to that previously described for char particles. However, soot particles in combustion processes are generally composed of small, isolated spheres, typically on the order of 30 nm in diameter, or, more generally, as open-structured aggregates of such spheres, as shown in Fig. 9.24. While aggregated particles may reach very large characteristic sizes (up to or even larger than 1 μm), the open structure of most aggregates allows gas molecules to freely diffuse to each individual sphere (known as a primary particle). As a result, the characteristic particle size for oxidation is of the order of the primary particle size, such that under most conditions the burning rate is kinetically controlled (Zone I combustion) and a single-film model of the particle combustion process is appropriate (i.e., the residence time of CO through the boundary layer is too short to allow oxidation to CO2). Soot can be oxidized by molecular oxygen, through the process described previously for pulverized coal char oxidation, but in combustion systems the dominant oxidation of soot is performed by radical species, especially the hydroxyl radical, OH. This discovery was made by Fenimore and Jones

200 nm

FIGURE 9.24 Transmission electron micrograph of soot particles collected from a laminar jet diffusion flame burning kerosene in air.

547

Combustion of Nonvolatile Fuels

[53] using a two-stage burner system in which the fuel-rich products (including soot) from a premixed flat flame were cooled and mixed with oxygen, hydrogen, and diluent gas before being burned in a second flat flame that was operated at stoichiometries ranging from fuel-lean to slightly fuel-rich. The importance of soot oxidation by OH was later confirmed by several investigators using both a two-stage burner [54] and laminar diffusion flames [55–57]. Oxygen atoms (O) contribute to soot oxidation, but their concentrations are always lower than that of OH in flames, so OH tends to dominate the oxidation process. The reaction probabilities for O and OH with soot particles have been measured by Roth and co-workers in a series of shock tube experiments [58–60]. They have found that both radicals react with soot particles with a collision efficiency of between 0.10 and 0.20. In contrast, the reaction probability with O2 is at least an order of magnitude lower [55]. Of course, at lower temperatures and sufficiently lean mixtures, soot oxidation by radical species becomes small and oxidation by O2 is important (though slow). Consequently, soot that passes through or avoids the primary reaction zone of a flame (e.g., due to local flame quenching) may experience oxidation from O2 in the postflame gases. Analysis of soot oxidation rates in flames [54–57] has supported the approximate value of the OH collision efficiency determined by Roth and co-workers. Unfortunately, OH and O concentrations in flames are determined by detailed chemical kinetics and cannot be accurately predicted from simple equilibrium at the local temperature and stoichiometry. This is particularly true when active soot oxidation is occurring and the local temperature is decreasing with flame residence time [59]. As a consequence, most attempts to model soot oxidation in flames have by necessity used a relation based on oxidation by O2 and then applied a correction factor to augment the rate to approximate the effect of oxidation by radicals. The two most commonly applied rate equations for soot oxidation by O2 are those developed by Lee et al. [61] and Nagle and Strickland-Constable [62]. Lee et al. were able to fit their data of soot burnout at the top of a flame by employing a simple first-order rate equation: w O2  1.085  105

PO2 T 0.5

⎛ 164.5 ⎞⎟ exp ⎜⎜ ⎟ ⎜⎝ RT ⎟⎠

(9.49)

where PO2 is given in atm, T is in Kelvin, and R has the units of kJ/(mol K). This may be further simplified to pure Arrhenius form as ⎛ 155.8 ⎞⎟ w O2  1352 PO2 exp ⎜⎜ ⎟ ⎜⎝ RT ⎟⎠

(9.50)

548

Combustion

Comparison of the Lee rate with the rate derived by Smith for a variety of carbon types [Eq. (9.48)] shows that the Lee rate is twice as high as the Smith rate at 1800 K. Nagle and Strickland-Constable (NSC) performed oxidation measurements on pyrocarbon rods subjected to high velocity oxidizer. For temperatures greater than 2000 K they measured a dip in the oxidation rate until the surface reached a temperature that was substantially greater and the oxidation rate once again increased with increasing temperature. To explain this phenomenon, they proposed that carbon oxidation proceeds through two types of surface sites, types A and B, that both react with oxygen to produce CO and generate a new type A site. Type A sites, meanwhile, thermally rearrange to type B sites. These presumptions yield an overall rate equation of the form ⎛ k P ⎜ A O2 w O2  ⎜⎜ ⎜⎜⎝ 1  k Z PO 2

⎞⎟ ⎟⎟ x  k P (1  x ) B O2 ⎟⎟ ⎠

(9.51)

where x

1 . 1  (kT /k B )PO2

With appropriate choices of kinetic constants, this approach can reproduce the NSC experimental data quite well. Park and Appleton [63] oxidized carbon black particles in a series of shock tube experiments and found a similar dependence of oxidation rate on oxygen concentration and temperature as NSC. Of course, the proper kinetic approach for soot oxidation by O2 undoubtedly should involve a complex surface reaction mechanism with distinct adsorption and desorption steps, in addition to site rearrangements, as suggested previously for char surface combustion.

PROBLEMS (Those with an asterisk require a numerical solution.) 1. Consider a spherical metal particle that is undergoing a high-temperature surface oxidation process. The product of this reaction is a nonvolatile oxide that immediately dissolves in the metal itself. The surface reaction and oxide dissolving rates are very fast compared to the oxidizer diffusion rate. Calculate an expression for the burning rate of this metal. 2. Calculate the value of the transfer number for silicon combustion in air. Show all the stoichiometric relationships in the calculation. 3. A carbon particle is large enough so that the burning rate is diffusioncontrolled. In one case the carbon monoxide leaving the surface burns to carbon dioxide in the gas phase; in another, no further carbon monoxide

Combustion of Nonvolatile Fuels

549

combustion takes place. Is the burning rate of the particle different in the two cases? If so, which is larger? Explain. 4.*The stoichiometric reaction between Al and liquid water is being considered as a means to generate hydrogen and for underwater propulsion. Determine whether the reaction will proceed heterogeneously or in the vapor phase at 20 atm. 5.*Determine the adiabatic flame temperature of a particle mixture containing 2 moles of aluminum and three moles of copper oxide (CuO(s)). Are the reaction products gaseous, liquid, or solids?

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26. Pivkina, A., Ulyanova, P., Frolov, Y., Zavyalov, S., and Schoonman, J. Prop. Explos. Pyrotech. 29, 39 (2004). 27. Beck, D. D., and Siegel, R. W., J. Mater. Res. 7, 2840 (1992). 28. Zachariah, M. R., Aquino, M. I., Shull, R. D., and Steel, E. B., Nanostruct. Mat. 5, 383 (1995). 29. Karch, J., and Birringer, R., Ceramics Intl 16, 291 (1990). 30. Eckert, J., Holzer, J. C., Ahn, C. C., Fu, Z., and Johnson, W. L., NanoStruct. Mat. 2, 407 (1993). 31. Puri, P., and Yang, V., J. Phys. Chem. 111, 11776 (2007). 32. Gregory, O. J., Lee, S. B., and Flagan, R. C., Comm. Am. Cer. Soc. 70, C-52 (1987). 33. Parr, T. P., Johnson, C., Hanson-Parr, D., Higa, K., and Wilson, K., 39th JANNAF Combustion Subcommittee Meeting, December 2003. 34. Bulian, C. J., Kerr, T. T., and Puszynski, J. A., 31st International Pyrotechnics Seminar, Fort Collins, CO, 12–14 July 2004, p. 327. 35. Son, S. F., Yetter, R. A., and Yang, V., eds., Special Issue on Nano Energetic Materials. J. Propul. Power 23, 4 (2007). 36. Glassman, I., Williams, F. A., and Antaki, P., Proc. Combust. Instit. 20, 2057 (1985). 37. King, M. K., Spacer. Rockets 19, 294 (1982). 38. Makino, A., Namikiri, T., and Kimura, K., Combust. Flame. 132, 274 (2007). 39. Solomon, P. R., “Coal Structure and Thermal Decomposition, in “New Approaches in Coal Chemistry,” ACS Symposium Series No. 169 (B. D. Blaustein, B. C. Bockrath, and S. Friedman, eds.), pp. 61–71. American Chemical Society, Washington, D.C., 1981. 40. Badzioch, S., Hawksley, P. G. W., and Peter, C. W., Ind. Eng. Chem. Process Des. Dev. 9, 521 (1970). 41. Solomon, P. R., Serio, M. A., and Suuberg, E. M., Prog. Energy Comb. Sci. 18, 133 (1992). 42. Kobayashi, H., Howard, J. B., and Sarofim, A. F., Proc. Comb. Instit. 16, 411 (1977). 43. Anthony, D. B., Howard, J. B., Hottel, H. C., and Meissner, H. P., Proc. Comb. Instit. 15, 1303 (1975). 44. Solomon, P. R., Hamblen, D. G., Serio, M. A., Yu, Z.-Z., and Charpenay, S., Fuel 72(4), 469 (1993). 45. Grant, D. M., Pugmire, R. J., Fletcher, T. H., and Kerstein, A. R., Energy & Fuels 3, 175 (1989). 46. Niksa, S., and Kerstein, A. R., Energy & Fuels 5, 647 (1991). 47. Thiele, E. W., Ind. Eng. Chem. 31, 916 (1939). 48. Hurt, R. H., and Calo, J. M., Combust. Flame. 125, 1138 (2001). 49. Smith, I. W., Fuel 57, 409 (1978). 50. Mitchell, R. E., Ma, L., and Kim, B., Combust. Flame 15, 426 (2007). 51. Williams, T. C., Shaddix, C. R., and Schefer, R. W., Combust. Sci. Technol. 180, 64–88 (2008). 52. NASA (2006), http://www.grc.nasa.gov/WWW/CEAWeb/. 53. Fenimore, C. P., and Jones, G. W., J. Phys. Chem. 71, 593 (1967). 54. Neoh, K. G., Howard, J. B., and Sarofim, A. F., in “Particulate Formation during Combustion.” (D. C. Siegla and G. W. Smith, eds.), pp. 261–282. Plenum, New York, 1981. 55. Garo, A., Prado, G., and Lahaye, J., Combust. Flame 79, 226 (1990). 56. Puri, R., Santoro, R. J., and Smyth, K. C., Combust. Flame 97, 125 (1994) and Erratum 102, 226 (1995). 57. Xu, F., El-Leathy, A. M., Kim, C. H., and Faeth, G. M., Combust. Flame 132, 43 (2003). 58. Roth, P., Brandt, O., and Von Gersum, S., Proc. Comb. Instit. 23, 1485 (1990). 59. Vierbaum, R., and Roth, P., Proc. Comb. Instit. 29, 2423 (2002). 60. Von Gersum, S., and Roth, P., Proc. Comb. Instit. 24, 999 (1922). 61. Lee, K. B., Thring, M. W., and Beer, J. M., Fuel 6, 137 (1962). 62. Nagle, J., Strickland-Constable, R. F., Proceedings of the Fifth Carbon Conference, Pergamon, Oxford, 1962, Vol. 1, pp. 154–164. 63. Park, C., and Appleton, J. P., Combust. Flame 20, 369 (1973).

Appendixes

The data presented in these appendixes are provided as a convenience to assist in solving some of the problems in the text, to meet preliminary research needs, to make rapid estimates in evaluating physical concepts, and to serve as a reference volume for the combustion community. Data in certain areas are constantly changing and thus those presented are subject to change. Although judgment was used in selecting the data and the most reliable and recent values available were chosen, it is important to note further that those presented are not the result of an extensive, critical survey. The reader is cautioned to use the material with care and to seek out original sources when a given datum value is crucial to a research result.

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CONTENTS Contents A.

Thermochemical Data and Conversion Factors Table 1. Conversion factors and physical constants Table 2. Thermochemical data for selected species Table 3. Thermochemical data for species included in reaction list of Appendix C

B.

Adiabatic Flame Temperatures of Hydrocarbons Table 1. Adiabatic flame temperatures

C.

Specific Reaction Rate Constants Table 1. Table 2. Table 3. Table 4. Table 5. Table 6. Table 7. Table 8. Table 9. Table 10. Table 11.

H2/O2 mechanism CO/H2/O2 mechanism CH2O/CO/H2/O2 mechanism CH3OH/CH2O/CO/H2/O2 mechanism CH4/CH3OH/CH2O/CO/H2/O2 mechanism C2H6/CH4/CH3OH/CH2O/CO/H2/O2 mechanism Selected reactions of a C3H8 oxidation mechanism NxOy/CO/H2/O2 mechanism HCl/NxOy/CO/H2/O2 mechanism O3/NxOy/CO/H2/O2 mechanism SOx/NxOy/CO/H2/O2 mechanism

D.

Bond Dissociation Energies of Hydrocarbons Table 1. Bond dissociation energies of alkanes Table 2. Bond dissociation energies of alkenes, alkynes, and aromatics Table 3. Bond dissociation energies of C/H/O compounds Table 4. Bond dissociation energies of sulfur-containing compounds Table 5. Bond dissociation energies of nitrogen-containing compounds Table 6. Bond dissociation of halocarbons

E.

Laminar Flame Speeds Table 1. Burning velocities of various fuels at 25oC air-fuel temperature (0.31 mol-%H2O in air). Burning velocity S as a function of equivalence ratio φ in cm s1 Table 2. Burning velocities of various fuels at 100oC air-fuel temperature (0.31 mol-%H2O in air). Burning velocity S as a function of equivalence ratio φ in cm s1 553

554

Combustion

Table 3. Burning velocities of various fuels in air as a function of pressure for an equivalence ratio of 1 in cm s1 F.

Flammability Limits in Air Table 1. Flammability limits of fuel gases and vapors in air at 25oC and 1 atm

G. Spontaneous Ignition Temperature Data Table 1. Spontaneous ignition temperature data H. Minimum Spark Ignition Energies and Quenching of Hydrocarbon Table 1. Minimum spark ignition energy data for fuels in air at 1 atm pressure I.

Programs for Combustion Kinetics

Appendix A

Thermochemical Data and Conversion Factors The thermochemical data for the chemical compounds that follow in this appendix are extracted directly from the JANAF tables [“JANAF thermochemical tables,” 3rd Ed., Chase, M. W., Jr., Davies, C. A., Davies, J. R., Jr., Fulrip, D. J., McDonald, R. A., and Syverud, A. N., J. Phys. Chem. Ref. Data 14, Suppl. 1 (1985)]. The compounds chosen from the numerous ones given are those believed to be most frequently used and those required to solve some of the problem sets given in Chapter 1. Since SI units have been used in the JANAF tables, these units were chosen as the standard throughout. Conversion to cgs units is readily accomplished by use of the conversion factors in this appendix (Table A1). Table A2 contains the thermochemical data. The ordered listing of the chemical compounds in Table A2 is the same as that in the JANAF tables and is alphabetical according to the chemical formula with the lowest order letter in the formula determining the position. The thermochemical tables have the following order:

Al (cr, l)

B2O3 (g)

HO2 (g)

SO (g)

Al (g)

Be (cr, l)

H2 (g)

TiO (g)

AlN (cr)

Be (g)

H2O (g)

O2 (g)

AlN (g)

BeO (cr, l)

NH3 (g)

SO2 (g)

AlO (g)

BeO (g)

Mg (cr, l)

TiO2 (cr, l)

Al2O (g)

C graphite

Mg (g)

TiO2 (g)

Al2O3 (cr, l)

C (g)

MgO (cr, l)

O3 (g)

B (cr, l)

CH4 (g)

MgO (g)

SO3 (g)

B (g)

CO (g)

N (g)

Ti3O5 (cr, l)

BHO2 (g)

CO2 (g)

NO (g)

Ti3O5 (l)

BO (g)

C2H2 (g)

NO (g)

S (cr, l)

BO2 (g)

C2H4 (g)

NO2 (g)

S (g)

B2O2 (g)

H (g)

N2 (g)

Ti (cr, l)

B2O3 (cr, l)

OH (g)

O (g)

Ti (g)

555

556

Combustion

The reader should refer to the original tables for the reference material on which the thermochemical data are based. The reference state used in Chapter 1 was chosen as 298 K; consequently, the thermochemical values at this temperature are identified from this listing. The logarithm of the equilibrium constant is to the base 10. The unit notation (J/K/mol) is equivalent to (JK1 mol1). Supplemental thermochemical data for species included in the reaction listing of Appendix C, and not given in Table A2, are listed in Table A3. These data, in combination with those of Table A2, may be used to calculate heats of reaction and reverse reaction rate constants as described in Chapter 2. References for the thermochemical data cited in Table A3 may be found in the respective references for the chemical mechanisms of Appendix C. Detailed data on the higher-order hydrocarbons are not presented. Such data are obtained readily from NBS Circular C 461, “Selected Values of Properties of Hydrocarbons,” 1947, from the work of Stull, D. R., Westrum, E. F., Jr., and Sinke, G. C., “The Chemical Thermodynamics of Organic Compounds,” John Wiley & Sons, New York, 1969, and from Reid, R. C., Prausnitz, J. M, and Poling, B. E., “The Properties of Gases and Liquids,” McGraw-Hill, New York, 1987. Burcat [“Thermochemical Data for Combustion Calculations,” in Combustion Chemistry. (W. C. Gardiner, Jr., ed.), Chapter 8. John Wiley & Sons, New York, 1984] discusses in detail the various sources of thermochemical data and their adaptation for computer usage. Examples of thermochemical data fit to polynomials for use in computer calculations are reported by McBride, B. J., Gordon, S., and Reno, M. A., “Coefficients for Calculating Thermodynamic and Transport Properties of Individual Species,” NASA, NASA Langley, VA, NASA Technical Memorandum 4513, 1993, and by Kee, R. J., Rupley, F. M., and Miller, J. A., “The Chemkin Thermodynamic Data Base,” Sandia National Laboratories, Livermore, CA, Sandia Technical Report SAND87-8215B, 1987. Thermochemical data are also available from the Internet. Some examples are the NIST Chemical Kinetics Model Database (http://kinetics.nist. gov/CKMech/), the Third Millennium Ideal Gas and Condensed Phase Thermochemical Database for Combustion (A. Burcat and B. Ruscic, ftp://ftp. technion.ac.il/pub/supported/aetdd/thermodynamics/), and the Sandia National Laboratory high-temperature thermodynamic database (http://www.ca.sandia. gov/HiTempThermo/). TABLE A1 Conversion Factors and Physical Constants 1 J  1 W s  1 N m  107 erg 1 cal (International Table)  4.1868 J 1 cal (Thermochemical)  4.184 J 1 cal/(g K)  1 kcal/(kg K)  4.1868 kJ/(kg K) 1 N  1 kg m/s2  105 dyne 1 Pa  1 N/m2 1 atm  1.0132  105 N/m2  1.0132  106 dyne/cm2

557

Appendix A: Thermochemical Data and Conversion Factors

TABLE A1 (continued) 1 bar  105 N/m2  105 Pa gs  Gravitational acceleration conversion factor  1 kg m/(N s2)  9.80665 m/s2 R  Universal gas constant  8.314 J/(g mol K)  1.987 cal/(g mol K)  82.05 cm3 atm/(g mol K) σ  Stefan–Boltzmann constant  5.6697  108 W/(m2 K4) kB  Boltzmann constant  1.38054  1016 erg/(molecule K) NA  Avogadro number 6.02252  1023 molecules/(g mol) h  Planck constant  6.256  1027 erg s c  Speed of light  2.997925  108 m/s

TABLE A2 Thermochemical Data of Selected Chemical Compounds Aluminum (Al), crystal–liquid, molecular weight  26.98154 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p (J/K/mol)

S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol) 4.539 4.056 2.254 1.140

0. 0. 0. 0.

ΔfG° (kJ/mol) 0. 0. 0. 0.

log Kf

0 100 200 250

0. 12.997 21.338 23.084

0. 6.987 19.144 24.108

Infinite 47.543 30.413 28.668

0. 0. 0. 0.

298.15

24.209

28.275

28.275

0.

0.

0.

0.

300 350 400 450 500

24.247 25.113 25.784 26.335 26.842

28.425 32.231 35.630 38.699 41.501

28.276 28.574 29.248 30.130 31.129

0.045 1.280 2.553 3.856 5.186

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

600 700 800 900

27.886 29.100 30.562 32.308

46.485 50.872 54.850 58.548

33.283 35.488 37.663 39.780

7.921 10.769 13.749 16.890

0. 0. 0. 0.

0. 0. 0. 0.

0. 0. 0. 0.

933.450 933.450

32.959 31.751

59.738 71.213

40.474 40.474

17.982 28.693

1000 1100 1200 1300 1400 1500

31.751 31.751 31.751 31.751 31.751 31.751

73.400 76.426 79.189 81.730 84.083 86.273

42.594 45.534 48.225 50.706 53.007 55.153

30.806 33.981 37.156 40.331 43.506 46.681

0. 0. 0. 0. 0. 0.

0. 0. 0. 0. 0. 0.

0. 0. 0. 0. 0. 0.

1600 1700 1800 1900 2000

31.751 31.751 31.751 31.751 31.751

88.323 90.247 92.062 93.779 95.408

57.162 59.052 60.836 62.525 64.129

49.856 53.031 56.207 59.382 62.557

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

Crystal–liquid transition

558

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Aluminum (Al), crystal–liquid, molecular weight  26.98154 Enthalpy reference temperature  Tr  298.15 K S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

T (K)

C°p (J/K/mol)

2100 2200 2300 2400 2500

31.751 31.751 31.751 31.751 31.751

96.957 98.434 99.845 101.196 102.493

65.656 67.112 68.505 69.839 71.120

65.732 68.907 72.082 75.257 78.432

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

2600 2700

31.751 31.751

103.738 104.936

72.350 73.535

81.607 84.782

0. 0.

0. 0.

0. 0.

2790.812

31.751

105.986

74.574

87.665

2800 2900 3000

31.751 31.751 31.751

106.091 107.205 108.281

74.677 75.780 76.846

87.957 91.132 94.307

log Kf

Fugacity  1 bar 293.901 292.805 291.710

0.968 11.479 21.953

0.018 0.207 0.382

TABLE A2 Thermochemical Data of Selected Chemical Compounds Aluminum (Al), ideal gas, molecular weight  26.98154 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p (J/K/mol)

S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

log Kf

0 100 200 250

0. 25.192 22.133 21.650

0. 139.619 155.883 160.764

Infinite 184.197 166.528 164.907

6.919 4.458 2.129 1.036

327.320 329.297 329.824 329.804

327.320 316.034 302.476 295.639

Infinite 165.079 78.999 61.770

298.15

21.390

164.553

164.553

0.

329.699

289.068

50.643

300 350 400 450 500

21.383 21.221 21.117 21.046 20.995

164.686 167.969 170.795 173.278 175.492

164.554 164.813 165.388 166.130 166.957

0.040 1.104 2.163 3.217 4.268

329.694 329.524 329.309 329.060 328.781

288.816 282.016 275.243 268.499 261.785

50.287 42.089 35.943 31.167 27.348

600 700 800 900 1000

20.930 20.891 20.866 20.849 20.836

179.314 182.537 185.325 187.781 189.977

168.708 170.460 172.147 173.751 175.266

6.363 8.454 10.542 12.628 14.712

328.141 327.385 326.492 325.436 313.605

248.444 235.219 222.112 209.126 197.027

21.629 17.552 14.502 12.137 10.292

1100 1200 1300

20.827 20.821 20.816

191.963 193.775 195.441

176.695 178.044 179.319

16.795 18.877 20.959

312.513 311.420 310.327

185.422 173.917 162.503

8.805 7.570 6.529

559

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Aluminum (Al), ideal gas, molecular weight  26.98154 Enthalpy reference temperature  Tr  298.15 K S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

180.526 181.672

23.041 25.122

309.233 308.139

151.172 139.921

5.640 4.872

199.762 201.023 202.212 203.337 204.404

182.761 183.798 184.789 185.736 186.643

27.202 29.283 31.363 33.443 35.523

307.045 305.950 304.856 303.760 302.665

128.742 117.631 106.585 95.600 84.673

4.203 3.614 3.093 2.628 2.211

20.797 20.796 20.795 20.795 20.794

205.419 206.386 207.311 208.196 209.044

187.513 188.349 189.153 189.928 190.676

37.603 39.682 41.762 43.841 45.921

301.570 300.475 299.379 298.284 297.188

73.800 62.979 52.209 41.486 30.808

1.836 1.495 1.186 0.903 0.644

2600 2700

20.794 20.794

209.860 210.645

191.398 192.097

48.000 50.080

296.092 294.997

20.175 9.583

0.405 0.185

2790.812

20.795

211.333

192.712

51.968

2800 2900 3000

20.795 20.796 20.798

211.401 212.131 212.836

192.773 193.428 194.063

52.159 54.239 56.318

0. 0. 0.

0. 0. 0.

0. 0. 0.

3100 3200 3300 3400 3500

20.800 20.804 20.808 20.815 20.823

213.518 214.178 214.818 215.440 216.043

194.680 195.279 195.861 196.428 196.980

58.398 60.478 62.559 64.640 66.722

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

3600 3700 3800 3900 4000

20.833 20.846 20.862 20.881 20.904

216.630 217.201 217.757 218.299 218.828

197.518 198.042 198.553 199.053 199.541

68.805 70.889 72.974 75.061 77.150

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

4100 4200 4300 4400 4500

20.932 20.964 21.002 21.046 21.088

219.345 219.849 220.343 220.826 221.299

200.017 200.484 200.940 201.386 201.823

79.242 81.337 83.435 85.537 87.642

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

4600 4700 4800 4900 5000

21.143 21.206 21.276 21.352 21.439

221.763 222.219 222.666 223.105 223.537

202.252 202.672 203.084 203.488 203.885

89.753 91.870 93.995 96.125 98.264

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

5100 5200 5300 5400 5500

21.535 21.641 21.757 21.884 22.021

223.963 224.382 224.795 225.203 225.606

204.274 204.657 205.033 205.403 205.766

100.413 102.572 104.742 106.924 109.119

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

T (K)

C°p (J/K/mol)

1400 1500

20.811 20.808

196.984 198.419

1600 1700 1800 1900 2000

20.805 20.803 20.801 20.800 20.798

2100 2200 2300 2400 2500

log Kf

Fugacity  1 bar

560

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Aluminum (Al), ideal gas, molecular weight  26.98154 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p (J/K/mol)

5600 5700 5800 5900 6000

22.170 22.330 22.496 22.680 22.836

S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol) 226.004 226.398 226.787 227.173 227.552

206.124 206.476 206.823 207.165 207.501

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol) 111.328 113.553 115.792 118.050 120.306

0. 0. 0. 0. 0.

ΔfG° (kJ/mol)

log Kf

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

TABLE A2 Thermochemical Data of Selected Chemical Compounds Aluminum nitride (AlN), crystal, molecular weight  40.98824 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa

T (K)

C°p (J/K/mol)

S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol) 3.871 3.711 2.463

312.980 314.756 316.764

ΔfG° (kJ/mol) 312.980 306.283 296.990

log Kf

0 100 200

0. 5.678 19.332

0. 2.164 10.267

Infinite 39.274 22.581

Infinite 159.986 77.566

298.15

30.097

20.142

20.142

0.

317.984

286.995

50.280

300 400 500

30.254 36.692 40.799

20.329 29.987 38.647

20.143 21.416 24.013

0.056 3.428 7.317

318.000 318.594 318.808

286.803 276.301 265.697

49.937 36.081 27.757

600 700 800 900 1000

43.538 45.434 46.791 47.792 48.550

46.341 53.201 59.361 64.932 70.008

27.106 30.353 33.601 38.777 39.850

11.541 15.994 20.608 25.339 30.158

318.811 318.727 318.648 318.647 329.363

255.072 244.455 233.850 223.252 211.887

22.206 18.241 15.269 12.957 11.068

1100 1200 1300 1400 1500

49.136 49.598 49.970 50.272 50.521

74.664 78.960 82.945 86.660 90.137

42.807 45.643 48.361 50.965 53.462

35.043 39.981 44.960 49.972 55.012

329.302 329.214 329.107 328.986 328.856

200.142 188.404 176.674 164.953 153.240

9.504 8.201 7.099 6.154 5.336

1600 1700 1800 1900 2000

50.728 50.903 51.052 51.180 51.290

93.404 96.485 99.399 102.162 104.790

55.857 58.157 60.368 62.496 64.545

60.075 65.157 70.255 75.368 80.490

328.717 328.573 328.425 328.273 328.119

141.537 129.843 118.157 106.480 94.810

4.621 3.990 3.429 2.927 2.476

561

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Aluminum nitride (AlN), crystal, molecular weight  40.98824 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

T (K)

C°p (J/K/mol)

2100 2200 2300 2400 2500

51.385 51.469 51.543 51.609 51.666

107.295 109.688 111.977 114.172 116.280

66.522 68.430 70.274 72.058 73.785

85.624 90.767 95.917 101.075 106.239

327.963 327.805 327.646 327.486 327.325

83.149 71.495 59.848 48.208 36.574

2.068 1.697 1.359 1.049 0.764

2600 2700 2800 2900 3000

51.718 51.765 51.807 51.845 51.878

118.308 120.260 122.144 123.962 125.721

75.458 77.082 78.658 80.189 81.677

111.408 116.582 121.761 126.943 132.130

327.165 327.003 620.743 619.486 618.229

24.948 13.327 0.744 21.376 43.454

0.501 0.258 0.014 0.385 0.757

log Kf

TABLE A2 Thermochemical Data of Selected Chemical Compounds Aluminum nitride (AlN), ideal gas, molecular weight  40.98824 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa [G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

C°p (J/K/mol)

S° (J/K/mol)

0 100 200 250

0. 29.134 30.316 31.378

0. 195.617 216.062 222.940

Infinite 255.951 231.448 229.080

8.942 6.033 3.077 1.535

522.933 523.906 523.605 523.306

522.933 513.034 502.220 496.908

Infinite 267.981 131.166 103.823

298.15

32.367

228.553

228.553

0.

523.000

491.851

86.170

300 350 400 450 500

32.403 33.288 34.019 34.612 35.093

228.753 233.817 238.311 242.353 246.026

228.553 228.951 229.846 231.015 232.335

0.060 1.703 3.386 5.102 6.845

522.988 522.668 522.348 522.028 521.704

491.658 486.462 481.311 476.201 471.126

85.605 72.600 62.853 55.276 49.218

600 700 800 900 1000

35.808 36.304 36.663 36.934 37.147

252.491 258.051 262.923 267.258 271.160

235.170 238.051 240.862 243.558 246.127

10.393 14.000 17.649 21.330 25.034

521.025 520.263 519.377 518.328 506.497

461.074 451.141 441.325 431.630 422.821

40.140 33.665 28.816 25.051 22.086

1100 1200 1300 1400 1500

37.320 37.465 37.589 37.699 37.797

274.709 277.963 280.967 283.756 286.361

248.566 250.882 253.082 255.175 257.168

28.758 32.497 36.250 40.014 43.789

505.396 504.286 503.167 502.040 500.905

414.507 406.293 398.172 390.138 382.184

19.683 17.685 15.999 14.556 13.309

T (K)

log Kf

562

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Aluminum nitride (AlN), crystal, molecular weight  40.98824 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa [G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

C°p (J/K/mol)

S° (J/K/mol)

1600 1700 1800 1900 2000

37.886 37.969 38.046 38.119 38.189

288.803 291.102 293.275 295.334 297.291

259.070 260.887 262.627 264.294 265.896

47.573 51.366 55.167 58.975 62.791

499.765 498.620 497.471 496.320 495.166

374.307 366.501 358.762 351.087 343.473

12.220 11.261 10.411 9.652 8.971

2100 2200 2300 2400 2500

38.257 38.322 38.387 38.450 38.514

299.156 300.937 302.642 304.277 305.848

267.435 268.918 270.347 271.727 273.061

66.613 70.442 74.278 78.119 81.968

494.011 492.855 491.698 490.542 489.387

335.917 328.416 320.967 313.569 306.219

8.355 7.798 7.289 6.825 6.398

2600 2700 2800 2900 3000

38.577 38.641 38.706 38.772 38.840

307.360 308.817 310.223 311.583 312.898

274.351 275.601 276.812 277.988 279.130

85.822 89.683 93.550 97.424 101.305

488.234 487.082 192.031 191.979 191.930

298.915 291.655 285.406 288.742 292.079

6.005 5.642 5.324 5.201 5.086

3100 3200 3300 3400 3500

38.910 38.981 39.056 39.133 39.212

314.173 315.409 316.610 317.777 318.913

280.240 281.320 282.371 283.395 284.394

105.192 109.087 112.989 116.898 120.815

191.884 191.842 191.804 191.770 191.741

295.418 298.759 302.101 305.444 308.787

4.978 4.877 4.782 4.693 4.608

3600 3700 3800 3900 4000

39.294 39.379 39.467 39.558 39.652

320.018 321.096 322.148 323.174 324.177

285.368 286.319 287.248 288.157 289.045

124.741 128.674 132.616 136.568 140.528

191.716 191.696 191.681 191.671 191.665

312.132 315.477 318.823 322.169 325.515

4.529 4.454 4.383 4.315 4.251

4100 4200 4300 4400 4500

39.748 39.847 39.949 40.053 40.159

325.157 326.116 327.055 327.974 328.876

289.913 290.764 291.597 292.413 293.214

144.498 148.478 152.468 156.468 160.478

191.665 191.670 191.679 191.693 191.713

328.861 332.207 335.553 338.899 342.244

4.190 4.132 4.076 4.023 3.973

4600 4700 4800 4900 5000

40.268 40.378 40.490 40.604 40.720

329.759 330.627 331.478 332.314 333.135

293.999 294.769 295.525 296.267 296.996

164.500 168.532 172.575 176.630 180.696

191.735 191.760 191.788 191.819 191.851

345.589 348.934 352.277 355.621 358.963

3.924 3.878 3.834 3.791 3.750

5100 5200 5300 5400 5500

40.836 40.954 41.073 41.192 41.313

333.943 334.737 335.518 336.287 337.044

297.713 298.417 299.110 299.791 300.462

184.774 188.864 192.965 197.078 201.203

191.884 191.917 191.948 191.978 192.005

362.305 365.646 368.987 372.327 375.667

3.711 3.673 3.637 3.602 3.568

5600 5700 5800 5900 6000

41.433 41.554 41.675 41.795 41.916

337.790 338.524 339.248 339.961 340.665

301.122 301.771 302.411 303.042 303.663

205.341 209.490 213.651 217.825 222.010

192.028 192.046 192.060 192.064 192.082

379.006 382.344 385.682 389.021 392.357

3.535 3.504 3.473 3.444 3.416

T (K)

log Kf

563

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Aluminum oxide (AlO), ideal gas, molecular weight  42.98094 Enthalpy reference temperature  Tr  298.15 K T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

0 100 200

0. 29.108 29.504

0. 186.129 206.370

Infinite 244.933 221.161

8.788 5.880 2.958

67.037 68.009 67.673

67.037 58.780 49.577

Infinite 30.693 12.948

298.15

30.883

218.386

218.386

0.

66.944

40.845

7.156

300 400 500

30.913 32.489 33.750

218.577 227.691 235.083

218.387 219.619 221.995

0.057 3.229 6.544

66.929 66.107 65.260

40.683 32.057 23.642

7.084 4.186 2.470

600 700 800 900 1000

34.685 35.399 36.001 36.583 37.205

241.323 246.726 251.492 255.766 259.652

224.710 227.478 230.187 232.796 235.290

9.968 13.474 17.044 20.673 24.362

64.369 63.400 62.321 61.106 49.149

15.401 7.315 0.625 8.422 15.315

1.341 0.546 0.041 0.489 0.800

1100 1200 1300 1400 1500

37.898 38.669 39.506 40.386 41.283

263.230 266.560 269.688 272.648 275.465

237.670 239.940 242.109 244.186 246.178

28.116 31.944 35.852 39.847 43.930

47.973 46.852 45.793 44.806 43.893

21.704 27.988 34.182 40.296 46.343

1.031 1.218 1.373 1.503 1.614

1600 1700 1800 1900 2000

42.167 43.015 43.805 44.523 45.158

278.157 280.739 283.221 285.609 287.909

248.093 249.938 251.719 253.440 255.106

48.103 52.362 56.704 61.121 65.606

43.057 42.296 41.605 40.977 40.405

52.331 58.269 64.164 70.023 75.850

1.708 1.790 1.862 1.925 1.981

2100 2200 2300 2400 2500

45.706 46.165 46.538 46.830 47.047

290.126 292.263 294.324 296.311 298.227

256.721 258.288 259.811 261.290 262.730

70.150 74.744 79.380 84.049 88.743

39.881 39.396 38.942 38.509 38.091

81.649 87.425 93.179 98.914 104.632

2.031 2.076 2.116 2.153 2.186

2600 2700 2800 2900 3000

47.196 47.286 47.325 47.321 47.279

300.075 301.858 303.579 305.240 306.843

264.131 265.495 266.825 268.121 269.385

93.456 98.180 102.911 107.644 112.374

37.681 37.272 257.042 256.368 255.706

110.333 116.018 120.719 115.863 111.029

2.217 2.244 2.252 2.087 1.933

3100 3200 3300 3400 3500

47.209 47.115 47.003 46.878 46.744

308.393 309.890 311.338 312.739 314.096

270.619 271.823 272.998 274.147 275.269

117.099 121.815 126.521 131.215 135.896

255.059 254.430 253.820 253.231 252.663

106.217 101.426 96.654 91.901 87.164

1.790 1.656 1.530 1.412 1.301

3600 3700

46.605 46.464

315.411 316.686

276.366 277.438

140.564 145.217

252.118 251.596

82.443 77.737

1.196 1.097

564

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Aluminum oxide (AlO), ideal gas, molecular weight  42.98094 Enthalpy reference temperature  Tr  298.15 K T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

3800 3900 4000

46.324 46.187 46.055

317.923 319.125 320.293

278.487 279.514 280.519

149.857 154.482 159.094

251.096 250.619 250.163

73.045 68.365 63.698

1.004 0.916 0.832

4100 4200 4300 4400 4500

45.930 45.813 45.704 45.606 45.518

321.428 322.534 323.610 324.660 325.684

281.503 282.467 283.411 284.337 285.244

163.694 168.281 172.857 177.422 181.978

249.730 249.318 248.928 248.557 248.205

59.042 54.396 49.760 45.133 40.514

0.752 0.677 0.604 0.536 0.470

4600 4700 4800 4900 5000

45.441 45.376 45.322 45.280 45.250

326.683 327.660 328.615 329.549 330.463

286.134 287.008 287.864 288.706 289.532

186.526 191.067 195.601 200.131 204.658

247.874 247.561 247.268 246.993 246.736

35.902 31.297 26.699 22.106 17.520

0.408 0.348 0.291 0.236 0.183

5100 5200 5300 5400 5500

45.232 45.226 45.231 45.248 45.276

331.359 332.237 333.099 333.944 334.775

290.343 291.140 291.924 292.694 293.452

209.182 213.705 218.227 222.751 227.277

246.498 246.278 246.077 245.894 245.730

12.937 8.360 3.787 0.783 5.349

0.133 0.084 0.037 0.008 0.051

5600 5700 5800 5900 6000

45.315 45.365 45.425 45.494 45.574

335.591 336.394 337.183 337.960 338.725

294.197 294.930 295.652 296.363 297.062

231.807 236.341 240.880 245.426 249.980

245.585 245.461 245.354 245.270 245.186

9.913 14.474 19.034 23.591 28.145

0.092 0.133 0.171 0.209 0.245

TABLE A2 Thermochemical Data of Selected Chemical Compounds Aluminum oxide (Al2O), ideal gas, molecular weight  69.96248 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa T (K)

C°p (J/K/mol)

S° [G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (J/K/mol) (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

0 100 200 250

0. 40.424 47.795 50.162

0. 201.813 232.403 243.332

Infinite 295.494 256.967 253.179

12.710 9.368 4.913 2.462

144.474 143.552 144.156 144.662

144.474 153.671 163.630 168.442

Infinite 80.269 42.736 35.194

298.15

52.032

252.332

252.332

0.

145.185

172.975

30.304

565

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Aluminum oxide (Al2O), ideal gas, molecular weight  69.96248 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa T (K)

C°p (J/K/mol)

S° [G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (J/K/mol) (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

300 350 400 450 500

52.098 53.696 55.010 56.087 56.971

252.654 260.809 268.068 274.611 280.568

252.333 252.973 254.415 256.301 258.435

0.096 2.742 5.461 8.239 11.067

145.205 145.767 146.342 146.929 147.532

173.147 177.760 182.291 186.750 191.142

30.148 26.529 23.805 21.677 19.968

600 700 800 900 1000

58.297 59.213 59.863 60.337 60.691

291.080 300.139 308.091 315.171 321.547

263.023 267.693 272.256 276.638 280.815

16.834 22.713 28.668 34.679 40.732

148.815 150.259 151.933 153.907 177.417

199.746 208.123 216.278 224.206 230.375

17.389 15.530 14.121 13.013 12.034

1100 1200 1300 1400 1500

60.961 61.172 61.339 61.473 61.583

327.345 332.658 337.562 342.112 346.357

284.786 288.557 292.140 295.549 298.796

46.815 52.922 59.048 65.188 71.341

179.438 181.456 183.472 185.488 187.505

235.573 240.587 245.433 250.124 254.670

11.186 10.472 9.862 9.332 8.868

1600 1700 1800 1900 2000

61.674 61.750 61.814 61.869 61.915

350.335 354.076 357.608 360.951 364.126

301.895 304.855 307.689 310.405 313.012

77.504 83.676 89.854 96.038 102.227

189.526 191.551 193.581 195.616 197.658

259.082 263.368 267.534 271.587 275.532

8.458 8.092 7.764 7.466 7.196

2100 2200 2300 2400 2500

61.956 61.991 62.022 62.050 62.075

367.148 370.031 372.787 375.427 377.961

315.519 317.931 320.257 322.501 324.669

108.421 114.618 120.819 127.023 133.229

199.708 201.764 203.829 205.902 207.984

279.376 283.121 286.773 290.335 293.810

6.949 6.722 6.513 6.319 6.139

2600 2700 2800 2900 3000

62.097 62.118 62.138 62.156 62.174

380.396 382.740 384.999 387.180 389.288

326.766 328.796 330.763 332.671 334.523

139.438 145.649 151.861 158.076 164.293

210.073 212.171 802.079 802.003 801.934

297.202 300.513 301.811 283.946 266.083

5.971 5.814 5.630 5.114 4.633

3100 3200 3300 3400 3500

62.192 62.210 62.229 62.248 62.269

391.327 393.301 395.216 397.074 398.879

336.323 338.073 339.776 341.434 343.049

170.511 176.731 182.953 189.177 195.403

801.873 801.820 801.775 801.737 801.707

248.222 230.363 212.506 194.650 176.795

4.183 3.760 3.364 2.990 2.639

3600 3700 3800 3900 4000

62.291 62.315 62.341 62.369 62.399

400.633 402.340 404.002 405.622 407.202

344.625 346.162 347.662 349.127 350.560

201.631 207.861 214.094 220.329 226.567

801.684 801.669 801.661 801.661 801.669

158.941 141.087 123.233 105.380 87.526

2.306 1.992 1.694 1.411 1.143

4100 4200 4300 4400 4500

62.433 62.470 62.509 62.553 62.600

408.743 410.248 411.718 413.156 414.562

351.960 353.330 354.671 355.984 357.270

232.809 239.054 245.303 251.556 257.814

801.685 801.710 801.744 801.788 801.839

69.673 51.818 33.963 16.108 1.749

0.888 0.644 0.413 0.191 0.020

566

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Aluminum oxide (Al2O), ideal gas, molecular weight  69.96248 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa S° [G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (J/K/mol) (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

T (K)

C°p (J/K/mol)

4600 4700 4800 4900 5000

62.651 62.705 62.764 62.827 62.895

415.938 417.286 418.607 419.902 421.172

358.530 359.766 360.978 362.168 363.335

264.076 270.344 276.617 282.897 289.183

801.904 801.983 802.075 802.180 802.303

19.607 37.467 55.328 73.192 91.058

0.223 0.416 0.602 0.780 0.951

5100 5200 5300 5400 5500

62.967 63.044 63.125 63.211 63.301

422.418 423.641 424.843 426.024 427.184

364.481 365.607 366.714 367.801 368.870

295.476 301.777 308.085 314.402 320.727

802.445 802.606 802.789 802.995 803.227

108.927 126.798 144.673 162.551 180.434

1.116 1.274 1.426 1.572 1.714

5600 5700 5800 5900 6000

63.396 63.496 63.601 63.710 63.823

428.326 429.449 430.554 431.642 432.714

369.922 370.956 371.974 372.977 373.963

327.062 333.407 339.761 346.127 352.503

803.487 803.776 804.092 804.448 804.796

198.321 216.213 234.110 252.012 269.918

1.850 1.981 2.108 2.231 2.350

log Kf

TABLE A2 Thermochemical Data of Selected Chemical Compounds Aluminum oxide (Al2O3), crystal–liquid, molecular weight  101.96128 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa T (K) 0 100 200

C°p (J/K/mol) 0. 12.855 51.120

S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol) 0. 4.295 24.880

Infinite 101.230 57.381

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

10.020 9.693 6.500

1663.608 1663.608 Infinite 1668.606 1641.642 857.506 1673.383 1612.656 421.183

298.15

79.015

50.950

50.950

0.

1675.692 1582.275 277.208

300 400 500

79.416 96.086 106.131

51.440 76.779 99.388

50.951 54.293 61.098

0.147 8.995 19.145

1675.717 1581.696 275.398 1676.342 1550.226 202.439 1676.045 1518.718 158.659

600 700 800 900 1000

112.545 116.926 120.135 122.662 124.771

119.345 137.041 152.873 167.174 180.210

69.177 77.632 86.065 94.296 102.245

30.101 41.586 53.447 65.591 77.965

1675.300 1674.391 1673.498 1672.744 1693.394

1100 1200

126.608 128.252

192.189 203.277

109.884 117.211

90.535 103.280

1692.437 1328.286 1691.366 1295.228

1487.319 129.483 1456.059 108.652 1424.931 93.038 1393.908 80.900 1361.437 71.114 63.075 56.380

567

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Aluminum oxide (Al2O3), crystal–liquid, molecular weight  101.96128 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa T (K)

C°p (J/K/mol)

S° [G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (J/K/mol) (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

1300 1400 1500

129.737 131.081 132.290

213.602 223.267 232.353

124.233 130.965 137.425

116.180 129.222 142.392

1690.190 1688.918 1687.561

1262.264 1229.393 1196.617

50.718 45.869 41.670

1600 1700 1800 1900 2000

133.361 134.306 135.143 135.896 136.608

240.925 249.039 256.740 264.067 271.056

143.628 149.592 155.333 160.864 166.201

155.675 169.060 182.533 196.085 209.710

1686.128 1684.632 1683.082 1681.489 1679.858

1163.934 1131.342 1098.841 1066.426 1034.096

37.999 34.762 31.888 29.318 27.008

2100 2200 2300

137.319 138.030 138.741

277.738 284.143 290.294

171.354 176.336 181.158

223.407 237.174 251.013

1678.190 1676.485 1674.743

1001.849 969.681 937.593

24.920 23.023 21.293

2327 2327

138.934 192.464

291.914 339.652

182.434 182.434

254.761 365.847

2400 2500

192.464 192.464

345.597 353.454

187.307 193.796

379.896 399.143

1557.989 1550.905

909.127 882.237

19.787 18.433

2600 2700 2800 2900 3000

192.464 192.464 192.464 192.464 192.464

361.002 368.266 375.265 382.019 388.544

200.083 206.179 212.093 217.837 223.419

418.389 437.636 456.882 476.128 495.375

1543.853 1536.832 2117.645 2108.494 2099.372

855.629 829.292 801.279 754.429 707.892

17.190 16.044 14.948 13.589 12.325

3100 3200 3300 3400 3500

192.464 192.464 192.464 192.464 192.464

394.855 400.965 406.888 412.633 418.212

228.848 234.132 239.277 244.292 249.182

514.621 533.868 553.114 572.360 591.607

2090.279 2081.214 2072.175 2063.162 2054.176

661.658 615.719 570.063 524.679 479.561

11.149 10.051 9.023 8.061 7.157

3600 3700 3800 3900 4000

192.464 192.464 192.464 192.464 192.464

423.634 428.908 434.040 439.040 443.912

253.953 258.610 263.160 267.606 271.953

610.853 630.100 649.346 668.592 687.839

2045.213 2036.276 2027.363 2018.474 2009.609

434.699 390.086 345.712 301.574 257.664

6.307 5.507 4.752 4.039 3.365

Alpha–liquid transition

568

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Boron (B), crystal–liquid, molecular weight  10.81 Enthalpy reference temperature  Tr  298.15 K T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

1.214 1.190 0.857 0.486

0. 0. 0. 0.

0. 0. 0. 0.

0. 0. 0. 0.

5.834

0.

0.

0.

0.

5.834 5.981 6.335 6.828 7.415

0.021 0.648 1.384 2.211 3.115

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

17.251 20.570 23.617 26.421 29.014

8.758 10.212 11.699 13.181 14.637

5.096 7.251 9.534 11.915 14.378

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

25.606 26.161 26.663 27.125 27.557

31.425 33.677 35.792 37.785 39.671

16.055 17.430 18.762 20.051 21.296

16.908 19.496 22.138 24.828 27.562

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

1600 1700 1800 1900 2000

27.966 28.356 28.732 29.097 29.452

41.463 43.170 44.801 46.365 47.866

22.501 23.667 24.796 25.891 26.952

30.338 33.155 36.009 38.901 41.828

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

2100 2200 2300

29.799 30.140 30.475

49.312 50.706 52.053

27.983 28.984 29.958

44.791 47.788 50.819

0. 0. 0.

0. 0. 0.

0. 0. 0.

2350 2350

30.641 31.750

52.710 74.075

30.435 30.435

52.346 102.554

2400 2500

31.750 31.750

74.744 76.040

31.351 33.113

104.142 107.317

0. 0.

0. 0.

0. 0.

2600 2700 2800 2900 3000

31.750 31.750 31.750 31.750 31.750

77.285 78.483 79.638 80.752 81.828

34.788 38.384 37.909 39.367 40.764

110.492 113.667 116.842 120.017 123.192

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

3100 3200 3300 3400 3500

31.750 31.750 31.750 31.750 31.750

82.869 83.877 84.854 85.802 86.723

42.106 43.398 44.637 45.834 46.989

126.367 129.542 132.717 135.892 139.067

0. 0. 0. 0. 0.

0 0 0. 0. 0.

0. 0. 0. 0. 0.

0 100 200 250

0. 1.076 5.998 8.821

0. 0.308 2.419 4.063

Infinite 12.207 6.704 6.007

298.15

11.315

5.834

300 350 400 450 500

11.405 13.654 15.693 17.361 18.722

5.904 7.834 9.794 11.742 13.644

600 700 800 900 1000

20.778 22.249 23.361 24.245 24.978

1100 1200 1300 1400 1500

Beta–liquid transition

569

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Boron (B), crystal–liquid, molecular weight  10.81 Enthalpy reference temperature  Tr  298.15 K [G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

T (K)

C°p (J/K/mol)

S° (J/K/mol)

3600 3700 3800 3900 4000

31.750 31.750 31.750 31.750 31.750

87.617 88.487 89.334 90.158 90.962

48.105 49.185 50.231 51.244 52.227

142.242 145.417 148.592 151.767 154.942

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

4100

31.750

91.746

53.181

158.117

0.

0.

0.

4139.449

31.750

92.050

53.550

159.369

4200 4300 4400 4500

31.750 31.750 31.750 31.750

92.511 93.258 93.988 94.702

54.109 55.010 55.888 56.743

161.292 164.467 167.642 170.817

log Kf

Fugacity  1 bar 479.849 478.782 477.677 476.593

7.024 18.604 30.158 41.687

0.087 0.226 0.358 0.484

TABLE A2 Thermochemical Data of Selected Chemical Compounds Boron (B), ideal gas, molecular weight  10.81 Enthalpy reference temperature  Tr  298.15 K T (K)

C°p S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol) 6.316 4.125 –2.042 1.001

0 100 200 250

0. 20.881 20.809 20.801

0. 130.687 145.130 149.772

Infinite 171.936 155.338 153.778

298.15

20.796

153.435

153.435

0.

300 350 400 450 500

20.796 20.793 20.792 20.790 20.790

153.564 156.769 159.546 161.995 164.185

153.436 153.689 154.251 154.978 155.791

0.038 1.078 2.118 3.157 4.197

600 700 800 900 1000

20.789 20.788 20.787 20.797 20.787

167.975 171.180 173.956 176.404 178.594

157.516 159.245 160.914 162.502 164.003

6.276 8.355 10.433 12.512 14.591

1100 1200 1300 1400 1500

20.797 20.787 20.787 20.786 20.786

180.575 182.384 184.048 185.588 187.023

165.421 166.761 168.027 169.227 170.366

16.669 18.748 20.827 22.905 24.984

554.898 557.065 558.815 559.485

ΔfG° (kJ/mol)

log Kf

554.898 544.027 530.273 523.057

Infinite 284.171 138.493 109.287

560.000

515.993

90.400

560.017 560.430 560.734 560.946 561.082

515.720 508.302 500.834 493.332 485.812

89.795 75.860 65.402 57.265 50.752

561.180 561.104 560.900 560.597 560.213

470.745 455.677 440.628 425.612 410.633

40.982 34.003 28.770 24.702 21.449

559.762 559.252 558.689 558.078 557.422

395.697 380.804 365.955 351.152 336.395

18.790 16.576 14.704 13.102 11.714

570

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Boron (B), ideal gas, molecular weight  10.81 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

1600 1700 1800 1900 2000

20.786 20.786 20.786 20.786 20.786

188.364 189.624 190.812 191.936 193.002

171.450 172.482 173.468 174.411 175.314

27.063 29.141 31.220 33.299 35.377

556.724 555.987 555.211 554.398 553.549

321.682 307.014 292.391 277.812 263.277

10.502 9.433 8.485 7.638 6.876

2100 2200 2300 2400 2500

20.786 20.786 20.786 20.786 20.786

194.017 194.984 195.908 196.792 197.641

176.180 177.013 177.815 178.587 179.333

37.456 39.535 41.613 43.692 45.770

552.665 551.747 550.795 499.550 498.453

248.785 234.335 219.929 206.633 194.451

6.188 5.564 4.995 4.497 4.063

2600 2700 2800 2900 3000

20.787 20.787 20.787 20.788 20.789

198.456 199.240 199.996 200.726 201.431

180.052 180.749 181.423 182.076 182.709

47.849 49.928 52.006 54.085 56.164

497.357 496.261 495.165 494.068 492.972

182.312 170.216 158.160 146.144 134.165

3.663 3.293 2.951 2.632 2.336

3100 3200 3300 3400 3500

20.791 20.793 20.795 20.798 20.803

202.112 202.772 203.412 204.033 204.636

183.324 183.922 184.503 185.068 185.619

58.243 60.322 62.402 64.481 66.561

491.876 490.780 489.685 488.589 487.494

122.223 110.316 98.444 86.604 74.797

2.059 1.801 1.558 1.331 1.116

3600 3700 3800 3900 4000

20.808 20.814 20.822 20.831 20.842

205.222 205.792 206.348 206.889 207.416

186.155 186.678 187.188 187.687 188.173

68.642 70.723 72.805 74.887 76.971

486.400 485.306 484.213 483.120 482.029

63.021 51.276 39.560 27.873 16.214

0.914 0.724 0.544 0.373 0.212

4100

20.855

207.931

188.649

79.056

480.939

4.582

0.058

4139.449

20.860

208.131

188.834

79.879

4200 4300 4400 4500

20.870 20.887 20.906 20.928

208.434 208.925 209.405 209.875

189.114 189.569 190.015 190.451

81.142 83.230 85.319 87.411

0. 0. 0. 0.

0. 0. 0. 0.

0. 0. 0. 0.

4600 4700 4800 4900 5000

20.952 20.979 21.009 21.042 21.078

210.336 210.786 211.228 211.662 212.087

190.878 191.297 191.707 192.110 192.506

89.505 91.602 93.701 95.803 97.909

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

5100 5200 5300 5400 5500

21.117 21.160 21.206 21.255 21.308

212.505 212.916 213.319 213.716 214.107

192.894 193.275 193.649 194.017 194.379

100.019 102.133 104.251 106.374 108.502

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

5600 5700 5800 5900 6000

21.365 21.425 21.489 21.557 21.628

214.491 214.870 215.243 215.611 215.974

194.735 195.084 195.429 195.768 196.101

110.636 112.776 114.921 117.073 119.233

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

Fugacity  1 bar

571

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Metaboric acid (BHO2), ideal gas, molecular weight  43.81674 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa

T (K) 0 100 200 250

C°p (J/K/mol)

S° (J/K/mol)

0. 33.404 36.887 39.583

0. 200.044 224.017 232.535

[G°  H°(Tr)]/T H°  H°(Tr) (J/K/mol) (kJ/mol) Infinite 273.638 243.422 240.414

10.687 7.359 3.881 1.970

ΔfH° (kJ/mol) 557.212 558.313 559.425 560.041

ΔfG° (kJ/mol) 557.212 555.919 553.107 551.456

log Kf Infinite 290.382 144.457 115.221

298.15

42.232

239.734

239.734

0.

560.656

549.748

96.313

300 350 400 450 500

42.332 45.007 47.553 49.941 52.157

239.995 246.723 252.901 258.641 264.019

239.735 240.260 241.459 243.053 244.883

0.078 2.262 4.577 7.015 9.568

560.680 561.324 561.968 562.605 563.228

549.680 547.796 545.819 543.762 541.635

95.708 81.754 71.277 63.118 56.584

600 700 800 900 1000

56.073 59.364 62.136 64.489 66.505

273.885 282.783 290.896 298.354 305.256

248.910 253.124 257.345 261.493 265.529

14.985 20.762 26.841 33.175 39.727

564.416 565.518 566.535 567.476 568.349

537.203 532.579 527.803 522.904 517.905

46.768 39.742 34.462 30.349 27.053

1100 1200 1300 1400 1500

68.244 69.752 71.066 72.215 73.223

311.678 317.682 323.318 328.628 333.645

269.436 273.209 276.849 280.359 283.746

46.467 53.368 60.411 67.576 74.849

569.168 569.944 570.686 571.406 572.113

512.820 507.663 502.443 497.166 491.839

24.352 22.098 20.188 18.549 17.127

1600 1700 1800 1900 2000

74.111 74.894 75.588 76.203 76.752

338.400 342.917 347.218 351.321 355.244

287.015 290.171 293.222 296.173 299.029

82.216 89.668 97.192 104.782 112.431

572.814 573.518 574.231 574.958 575.705

486.464 481.046 475.586 470.086 464.547

15.881 14.781 13.801 12.924 12.133

2100 2200 2300 2400 2500

77.242 77.681 78.075 78.430 78.751

359.001 362.605 366.067 369.397 372.605

301.796 304.479 307.082 309.609 312.065

120.131 127.877 135.666 143.491 151.350

576.475 577.273 578.102 629.224 630.199

458.970 453.356 447.705 440.949 433.084

11.416 10.764 10.168 9.597 9.049

2600 2700 2800 2900 3000

79.041 79.305 79.545 79.764 79.964

375.700 378.688 381.576 384.372 387.079

314.454 316.778 319.041 321.245 323.395

159.240 167.158 175.100 183.066 191.053

631.179 632.166 633.161 634.165 635.179

425.180 417.238 409.259 401.246 393.196

8.542 8.072 7.635 7.227 6.846

3100 3200 3300 3400 3500

80.148 80.316 80.472 80.615 80.747

389.704 392.252 394.725 397.130 399.469

325.492 327.539 329.537 331.490 333.399

199.058 207.082 215.121 223.176 231.244

636.204 637.240 638.288 639.349 640.422

385.113 376.997 368.849 360.668 352.456

6.489 6.154 5.838 5.541 5.260

3600 3700 3800 3900 4000

80.869 80.982 81.088 81.185 81.277

401.745 403.962 406.123 408.231 410.288

335.266 337.093 338.881 340.632 342.348

239.325 247.417 255.521 263.635 271.758

641.508 642.607 643.719 644.844 645.982

344.213 335.940 327.636 319.303 310.942

4.994 4.743 4.504 4.277 4.060

572

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Metaboric acid (BHO2), ideal gas, molecular weight  43.81674 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T H°  H°(Tr) (J/K/mol) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

4100 4200 4300 4400 4500

81.362 81.441 81.516 81.585 81.651

412.296 414.257 416.174 418.049 419.883

344.030 345.679 347.296 348.883 350.440

279.890 288.030 296.178 304.333 312.495

647.134 1128.149 1128.242 1128.350 1128.474

302.551 287.109 267.082 247.055 227.025

3.855 3.571 3.244 2.933 2.635

4600 4700 4800 4900 5000

81.712 81.770 81.824 81.876 81.924

421.679 423.437 425.159 426.846 428.501

351.969 353.471 354.947 356.397 357.823

320.663 328.837 337.017 345.202 353.392

1128.615 1128.774 1128.950 1129.146 1129.360

206.991 186.954 166.913 146.868 126.819

2.350 2.078 1.816 1.566 1.325

5100 5200 5300 5400 5500

81.970 82.013 82.054 82.092 82.129

430.124 431.716 433.279 434.813 436.319

359.225 360.603 361.960 363.295 364.609

361.586 369.786 377.989 386.196 394.407

1129.595 1129.850 1130.127 1130.427 1130.750

106.766 86.709 66.645 46.577 26.504

1.094 0.871 0.657 0.451 0.252

5600 5700 5800 5900 6000

82.164 82.197 82.229 82.258 82.287

437.799 439.254 440.684 442.090 443.473

365.903 367.177 366.432 369.668 370.887

402.622 410.840 419.061 427.286 435.513

1131.098 1131.472 1131.872 1132.299 1132.756

6.422 13.664 33.758 53.857 73.966

0.060 0.125 0.304 0.477 0.644

log Kf

TABLE A2 Thermochemical Data of Selected Chemical Compounds Boron monoxide (BO), ideal gas, molecular weight  26.8094 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol) 8.674 5.771 2.860 1.404

3.118 1.692 0.569 0.213

ΔfG° (kJ/mol) 3.118 10.162 19.105 23.783

log Kf

0 100 200 250

0. 29.105 29.112 29.133

0. 171.662 191.838 198.336

Infinite 229.370 206.138 203.952

Infinite 5.308 4.990 4.969

298.15

29.196

203.472

203.472

0.

0.

28.343

4.966

300 350 400 450 500

29.200 29.343 29.574 29.882 30.250

203.652 208.163 212.096 215.596 218.763

203.472 203.828 204.621 205.649 206.805

0.054 1.517 2.990 4.476 5.979

0.006 0.104 0.094 0.007 0.178

28.519 33.283 38.053 42.816 47.564

4.966 4.967 4.969 4.970 4.969

600 700

31.076 31.910

224.350 229.204

209.276 211.783

9.045 12.194

0.673 1.306

56.997 66.336

4.962 4.950

573

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Boron monoxide (BO), ideal gas, molecular weight  26.8094 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

800 900 1000

32.677 33.350 33.926

233.516 237.404 240.949

214.235 216.597 218.858

15.425 18.727 22.091

2.027 2.809 3.638

75.578 84.725 93.783

4.935 4.917 4.899

1100 1200 1300 1400 1500

34.415 34.829 35.181 35.481 35.739

244.206 247.219 250.021 252.639 255.096

221.016 223.075 225.042 226.920 228.718

25.509 28.972 32.473 36.006 39.568

4.505 5.405 6.337 7.300 8.294

102.756 111.648 120.464 129.208 137.881

4.879 4.860 4.840 4.821 4.801

1600 1700 1800 1900 2000

35.961 36.155 36.324 36.473 36.606

257.410 259.596 261.667 263.635 265.510

230.439 232.091 233.677 235.202 236.671

43.153 46.759 50.383 54.023 57.677

9.318 10.375 11.463 12.584 13.739

146.487 155.028 163.505 171.922 180.278

4.782 4.763 4.745 4.726 4.708

2100 2200 2300 2400 2500

36.724 36.831 36.928 37.016 37.098

267.299 269.010 270.649 272.222 273.735

238.087 239.454 240.775 242.053 243.290

61.344 65.022 68.710 72.407 76.113

14.928 16.151 17.409 68.961 70.368

188.576 196.816 205.000 212.060 217.194

4.691 4.673 4.656 4.615 4.555

2600 2700 2800 2900 3000

37.173 37.244 37.310 37.374 37.435

275.192 276.596 277.952 279.262 280.530

244.489 245.653 246.782 247.880 248.947

79.826 83.547 87.275 91.009 94.749

71.778 73.190 74.607 76.026 77.449

223.871 229.694 235.464 241.185 246.855

4.498 4.444 4.393 4.344 4.298

3100 3200 3300 3400 3500

37.495 37.554 37.613 37.672 37.731

281.759 282.950 284.106 285.230 286.323

249.986 250.997 251.983 252.945 253.883

98.496 102.248 106.007 109.771 113.541

78.876 80.305 81.737 83.172 84.609

252.478 258.056 263.589 269.078 274.525

4.254 4.212 4.172 4.134 4.097

3600 3700 3800 3900 4000

37.792 37.855 37.920 37.987 38.057

287.387 288.423 289.433 290.419 291.382

254.799 255.693 256.568 257.424 258.261

117.317 121.100 124.888 128.684 132.486

86.047 87.487 88.927 90.368 91.809

279.930 285.296 290.623 295.912 301.164

4.062 4.028 3.995 3.963 3.933

4100 4200 4300 4400 4500

38.130 38.206 38.285 38.368 38.455

292.323 293.242 294.142 295.023 295.887

259.080 259.882 260.669 261.439 262.195

136.295 140.112 143.936 147.769 151.610

93.249 574.537 574.887 575.237 575.586

306.380 304.537 298.104 291.663 285.215

3.903 3.787 3.621 3.462 3.311

4600 4700 4800 4900 5000

38.546 38.640 38.738 38.840 38.946

296.733 297.563 298.377 299.177 299.963

262.937 263.665 264.380 265.082 265.771

155.460 159.319 163.188 167.067 170.956

575.935 576.284 576.632 576.980 577.327

278.758 272.294 265.823 259.343 252.858

3.165 3.026 2.893 2.765 2.642

574

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Boron monoxide (BO), ideal gas, molecular weight  26.8094 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

5100 5200 5300 5400 5500

39.055 39.169 39.285 39.406 39.529

300.735 301.495 302.242 302.977 303.701

266.449 267.116 267.772 268.417 269.052

174.856 178.768 182.690 186.625 190.571

577.674 578.020 578.368 578.715 579.064

246.365 239.865 233.359 226.846 220.327

2.523 2.409 2.300 2.194 2.092

5600 5700 5800 5900 6000

39.656 39.787 39.920 40.055 40.194

304.415 305.118 305.811 306.494 307.169

269.677 270.293 270.899 271.497 272.086

194.531 198.503 202.488 206.487 210.499

579.414 579.766 580.120 580.477 580.838

213.801 207.269 200.731 194.187 187.637

1.994 1.899 1.808 1.719 1.634

TABLE A2 Thermochemical Data of Selected Chemical Compounds Boron dioxide (BO2), ideal gas, molecular weight  42.8088 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

10.731 7.620 4.019 2.030

285.346 285.163 284.807 284.646

285.346 286.678 288.331 289.231

Infinite 149.745 75.304 60.432

229.814

0.

284.512

290.127

50.829

229.815 230.351 231.567 233.174 235.007

0.080 2.303 4.630 7.049 9.550

284.507 284.388 284.292 284.217 284.160

290.162 291.114 292.082 293.060 294.046

50.522 43.446 38.142 34.018 30.719

263.597 271.953 279.406 286.123 292.228

238.999 243.122 247.200 251.158 254.964

14.759 20.182 25.765 31.469 37.264

284.093 284.080 284.116 284.200 284.329

296.030 298.021 300.011 301.994 303.964

25.772 22.239 19.589 17.527 15.877

58.957 59.451 59.847 60.169 60.435

297.818 302.970 307.745 312.192 316.353

258.610 262.094 265.425 268.608 271.654

43.129 49.051 55.016 61.018 67.048

284.502 284.719 284.978 285.280 285.624

305.920 307.857 309.776 311.672 313.546

14.527 13.401 12.447 11.629 10.919

60.655 60.841

320.260 323.943

274.571 277.368

73.103 79.178

286.013 286.446

315.395 317.218

10.297 9.747

0 100 200 250

0. 33.725 38.528 41.020

0. 188.764 213.526 222.394

Infinite 264.960 233.623 230.514

298.15

43.276

229.814

300 350 400 450 500

43.359 45.527 47.495 49.246 50.780

230.082 236.931 243.141 248.838 254.108

600 700 800 900 1000

53.263 55.110 56.490 57.533 58.333

1100 1200 1300 1400 1500 1600 1700

575

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Boron dioxide (BO2), ideal gas, molecular weight  42.8088 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

1800 1900 2000

60.998 61.133 61.250

327.425 330.727 333.866

280.053 282.634 285.118

85.270 91.377 97.496

286.924 287.449 288.019

319.015 320.783 322.523

9.258 8.819 8.423

2100 2200 2300 2400 2500

61.352 61.443 61.523 61.597 61.664

336.857 339.713 342.446 345.066 347.582

287.511 289.819 292.048 294.203 296.288

103.627 109.766 115.915 122.071 128.234

288.637 289.302 290.016 341.036 341.923

324.233 325.912 327.561 328.109 327.552

8.065 7.738 7.439 7.141 6.844

2600 2700 2800 2900 3000

61.728 61.788 61.847 61.904 61.961

350.002 352.332 354.581 356.752 358.852

298.308 300.266 302.166 304.011 305.804

134.404 140.579 146.761 152.949 159.142

342.824 343.741 344.672 345.617 346.575

326.959 326.332 325.670 324.975 324.247

6.569 6.313 6.075 5.853 5.646

3100 3200 3300 3400 3500

62.019 62.078 62.138 62.201 62.265

360.884 362.854 364.765 366.621 368.425

307.548 309.246 310.900 312.511 314.083

165.341 171.546 177.757 183.974 190.197

347.547 348.531 349.526 350.532 351.548

323.486 322.695 321.872 321.019 320.136

5.451 5.267 5.095 4.932 4.778

3600 3700 3800 3900 4000

62.333 62.403 62.475 62.551 62.629

370.180 371.889 373.554 375.178 376.763

315.617 317.115 318.579 320.009 321.408

196.427 202.663 208.907 215.159 221.418

352.572 353.604 354.644 355.690 356.741

319.224 318.284 317.315 316.318 315.296

4.632 4.493 4.362 4.237 4.117

4100 4200 4300 4400 4500

62.711 62.795 62.882 62.971 63.062

378.310 379.822 381.301 382.747 384.164

322.777 324.117 325.430 326.716 327.977

227.685 233.960 240.244 246.536 252.838

357.798 838.708 838.686 838.669 838.657

314.247 306.148 293.467 280.789 268.110

4.004 3.808 3.565 3.333 3.112

4600 4700 4800 4900 5000

63.156 63.252 63.350 63.449 63.549

385.551 386.910 388.243 389.550 390.833

329.214 330.427 331.618 332.787 333.935

259.149 265.469 271.799 278.139 284.489

838.650 838.649 838.653 838.664 838.680

255.432 242.753 230.074 217.395 204.715

2.901 2.698 2.504 2.317 2.139

5100 5200 5300 5400 5500

63.651 63.754 63.857 63.961 64.065

392.092 393.329 394.545 395.739 396.914

335.063 336.172 337.262 338.333 339.388

290.849 297.219 303.600 309.991 316.392

838.705 838.736 838.777 838.828 838.889

192.036 179.357 166.675 153.994 141.312

1.967 1.802 1.643 1.490 1.342

5600 5700 5800 5900 6000

64.169 64.273 64.377 64.480 64.582

398.069 399.206 400.324 401.426 402.510

340.425 341.447 342.452 343.443 344.418

322.804 329.226 335.658 342.101 348.554

838.962 839.048 839.149 839.266 839.400

128.627 115.942 103.254 90.567 77.875

1.200 1.062 0.930 0.802 0.678

576

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Boron monoxide, dimeric (BO)2, ideal gas, molecular weight  53.6188 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa

T (K)

C°p S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol) (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

12.398 9.367 5.220 2.671

457.343 457.265 456.694 456.345

457.343 459.202 461.326 462.525

Infinite 239.863 120.486 96.639

242.597

0.

456.056

463.743

81.246

242.598 243.309 244.923 247.053 249.479

0.106 3.056 6.141 9.337 12.629

456.046 455.827 455.708 455.684 455.741

463.791 465.100 466.434 467.777 469.118

80.753 69.412 60.910 54.298 49.008

287.184 298.147 307.962 316.853 324.980

254.750 260.182 265.551 270.765 275.786

19.461 26.576 33.928 41.479 49.194

456.031 456.480 457.030 457.649 458.320

471.770 474.359 476.877 479.321 481.694

41.071 35.397 31.137 27.819 25.161

79.116 80.149 81.013 81.739 82.353

332.462 339.392 345.842 351.873 357.534

280.603 285.217 289.635 293.868 297.925

57.045 65.010 73.069 81.208 89.413

459.038 459.800 460.607 461.461 462.365

483.996 486.232 488.402 490.509 492.552

22.983 21.165 19.624 18.301 17.152

1600 1700 1800 1900 2000

82.876 83.323 83.709 84.042 84.333

362.866 367.904 372.678 377.213 381.532

301.819 305.560 309.157 312.621 315.959

97.675 105.986 114.338 122.726 131.145

463.323 464.337 465.410 466.544 467.743

494.534 496.454 498.313 500.110 501.846

16.145 15.254 14.461 13.749 13.107

2100 2200 2300 2400 2500

84.588 84.812 85.010 85.185 85.342

385.653 389.593 393.367 396.989 400.470

319.180 322.292 325.301 328.213 331.034

139.591 148.062 156.553 165.063 173.589

469.007 470.339 471.740 573.730 575.428

503.520 505.132 506.683 506.034 503.178

12.524 11.993 11.507 11.014 10.513

2600 2700 2800 2900 3000

85.482 85.608 85.722 85.824 85.917

403.820 407.048 410.164 413.174 416.085

333.770 336.424 339.002 341.508 343.946

182.131 190.685 199.252 207.829 216.416

577.133 578.846 580.567 582.297 584.037

500.254 497.265 494.211 491.098 487.923

10.050 9.620 9.220 8.846 8.495

3100 3200 3300 3400 3500

86.002 86.079 86.150 86.214 86.274

418.903 421.635 424.285 426.858 429.358

346.319 348.630 350.883 353.079 355.223

225.012 233.617 242.228 250.846 259.471

585.787 587.546 589.316 591.095 592.885

484.690 481.401 478.057 474.658 471.208

8.167 7.858 7.567 7.292 7.032

3600 3700 3800 3900 4000

86.328 86.379 86.426 86.469 86.509

431.789 434.155 436.459 438.705 440.894

357.317 359.361 361.360 363.315 365.227

268.101 276.736 285.377 294.021 302.670

594.684 596.492 598.311 600.138 601.975

467.706 464.154 460.552 456.903 453.208

6.786 6.553 6.331 6.120 5.918

0 100 200 250

0 34.367 48.231 53.477

0. 193.295 221.483 232.836

Infinite 286.969 247.582 243.520

298.15

57.297

242.597

300 350 400 450 500

57.424 60.445 62.871 64.921 66.722

242.952 252.040 260.275 267.802 274.737

600 700 800 900 1000

69.820 72.409 74.575 76.378 77.874

1100 1200 1300 1400 1500

577

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Boron monoxide, dimeric (BO)2, ideal gas, molecular weight  53.6188 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa

T (K)

C°p S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol) (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

4100 4200 4300 4400 4500

86.547 86.581 86.614 86.644 86.672

443.031 445.117 447.155 449.146 451.094

367.099 368.931 370.727 372.487 374.212

311.323 319.979 328.639 337.302 345.968

603.820 1565.374 1565.063 1564.766 1564.481

449.466 431.630 404.640 377.657 350.681

5.726 5.368 4.915 4.483 4.071

4600 4700 4800 4900 5000

86.699 86.724 86.747 86.769 86.790

452.999 454.864 456.690 458.479 460.232

375.904 377.564 379.194 380.794 382.365

354.637 363.308 371.981 380.657 389.335

1564.211 1563.955 1563.715 1563.492 1563.287

323.711 296.746 269.787 242.831 215.881

3.676 3.298 2.936 2.589 2.255

5100 5200 5300 5400 5500

86.810 86.828 86.846 86.862 86.878

461.951 463.637 465.291 466.914 468.508

383.909 385.426 386.917 388.384 389.826

398.015 406.697 415.381 424.066 432.753

1563.101 1562.935 1562.791 1562.670 1562.573

188.935 161.993 135.052 108.116 81.181

1.935 1.627 1.331 1.046 0.771

5600 5700 5800 5900 6000

86.893 86.907 86.920 86.933 86.945

470.074 471.612 473.124 474.609 476.071

391.245 392.641 394.016 395.370 396.702

441.442 450.132 458.823 467.516 476.210

1562.503 1562.461 1562.449 1562.468 1562.521

54.246 27.314 0.381 26.550 53.484

0.506 0.250 0.003 0.235 0.466

TABLE A2 Thermochemical Data of Selected Chemical Compounds Boron oxide (B2O3), crystal–liquid, molecular weight  69.6182 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa

T (K)

C°p S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol) (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

9.293 8.537 5.245

1265.776 1269.425 1271.165

1265.776 Infinite 1244.466 650.042 1218.726 318.299

53.953

0.

1271.936

1192.796 208.973

53.954 56.614 62.105

0.116 7.194 15.582

1271.944 1272.047 1271.710

1192.305 207.599 1165.729 152.229 1139.180 119.009

110.360 126.039

68.744 75.825

24.969 35.149

1271.024 1270.036

1112.733 1086.426

129.464 162.757

77.477 77.477

37.587 61.657

0 100 200

0. 20.740 43.844

0. 10.983 32.869

Infinite 96.353 59.093

298.15

62.588

53.953

300 400 500

62.927 77.948 89.287

54.341 74.600 93.269

600 700

98.115 105.228

723 723

106.692 129.704

96.672 81.070

Crystal–liquid transition

578

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Boron oxide (B2O3), crystal–liquid, molecular weight  69.6182 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa

T (K)

C°p S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol) (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

800 900 1000

129.704 129.704 129.704

175.883 191.160 204.826

86.328 97.144 107.241

71.644 84.615 97.585

1243.112 1240.014 1237.160

1062.927 1040.593 1018.590

69.402 60.394 53.206

1100 1200 1300 1400 1500

129.704 129.704 129.704 129.704 129.704

217.188 228.474 238.855 248.468 257.416

116.683 125.535 133.858 141.706 149.125

110.556 123.526 136.496 149.467 162.437

1234.514 1232.044 1229.732 1227.561 1225.521

996.863 975.368 954.074 932.951 911.980

47.337 42.457 38.335 34.809 31.758

1600 1700 1800 1900 2000

129.704 129.704 129.704 129.704 129.704

265.787 273.650 281.064 288.077 294.730

156.157 162.840 169.204 175.277 181.085

175.408 188.378 201.348 214.319 227.289

1223.604 1221.804 1220.116 1218.538 1217.066

891.141 870.417 849.797 829.267 808.818

29.093 26.745 24.660 22.798 21.124

2100 2200 2300 2400 2500

129.704 129.704 129.704 129.704 129.704

301.058 307.092 312.857 318.378 323.672

186.649 191.987 197.118 202.056 206.816

240.260 253.230 266.200 279.171 292.141

1215.699 1214.435 1213.272 1312.728 1311.920

788.439 768.123 747.863 725.514 701.063

19.611 18.238 16.985 15.790 14.648

2600 2700 2800 2900 3000

129.704 129.704 129.704 129.704 129.704

328.759 333.654 338.372 342.923 347.320

211.409 215.846 220.139 224.294 228.322

305.112 318.082 331.052 344.023 356.993

1311.144 1310.400 1309.686 1309.002 1308.347

676.644 652.254 627.891 603.555 579.240

13.594 12.619 11.713 10.871 10.085

TABLE A2 Thermochemical Data of Selected Chemical Compounds Boron oxide (B2O3), ideal gas, molecular weight  69.6182 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

14.337 10.797 6.037 3.102

834.847 835.713 835.985 835.979

834.847 832.354 828.844 827.059

Infinite 434.777 216.472 172.805

283.768

0.

835.963

825.343

144.597

283.789 284.601 286.499

0.124 3.584 7.235

835.963 835.972 836.033

825.277 823.496 821.710

143.694 122.900 107.304

0 100 200 250

0. 40.042 55.330 61.832

0. 226.986 259.361 272.432

Infinite 334.961 289.548 284.842

298.15

66.855

283.768

300 350 400

67.028 71.228 74.727

284.182 294.841 304.587

579

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Boron oxide (B2O3), ideal gas, molecular weight  69.6182 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

450 500

77.728 80.362

313.566 321.894

289.014 291.891

11.048 15.002

836.152 836.317

819.913 818.100

95.173 85.466

600 700 800 900 1000

84.817 88.439 91.403 93.830 95.822

336.954 350.310 362.320 373.230 383.223

298.173 304.685 311.151 317.452 323.537

23.269 31.938 40.935 50.200 59.686

836.752 837.275 837.849 838.455 839.087

814.418 810.655 806.814 802.898 798.914

70.901 60.492 52.680 46.599 41.731

1100 1200 1300 1400 1500

97.462 98.819 99.950 100.897 101.696

392.435 400.975 408.931 416.374 423.363

329.387 335.001 340.386 345.550 350.507

69.353 79.169 89.109 99.153 109.284

839.743 840.428 841.146 841.902 842.701

794.864 790.755 786.586 782.361 778.081

37.745 34.421 31.605 29.190 27.095

1600 1700 1800 1900 2000

102.375 102.955 103.454 103.886 104.262

429.949 436.173 442.072 447.677 453.016

355.269 359.846 364.252 368.496 372.590

119.488 129.755 140.076 150.444 160.852

843.550 844.453 845.415 846.440 847.531

773.746 769.355 764.911 760.411 755.855

25.260 23.639 22.197 20.905 19.741

2100 2200 2300 2400 2500

104.591 104.880 105.136 105.363 105.565

458.111 462.983 467.651 472.131 476.436

376.542 380.361 384.056 387.633 391.099

171.295 181.769 192.270 202.795 213.342

848.691 849.924 851.230 953.131 954.747

751.243 746.573 741.847 734.924 725.799

18.686 17.726 16.848 15.995 15.165

2600 2700 2800 2900 3000

105.746 105.908 106.054 106.186 106.306

480.580 484.574 488.428 492.152 495.754

394.462 397.726 400.897 403.980 406.979

223.907 234.490 245.088 255.700 266.325

956.376 958.019 959.677 961.351 963.042

716.609 707.356 698.041 688.669 679.237

14.397 13.685 13.022 12.404 11.827

3100 3200 3300 3400 3500

106.415 106.515 106.606 106.689 106.765

499.242 502.622 505.901 509.085 512.178

409.899 412.744 415.518 418.223 420.864

276.961 287.608 298.264 308.929 319.602

964.749 966.474 968.214 969.971 971.744

669.748 660.205 650.608 640.956 631.254

11.285 10.777 10.298 9.847 9.421

3600 3700 3800 3900 4000

106.836 106.901 106.961 107.017 107.069

515.187 518.115 520.967 523.746 526.456

423.442 425.961 428.424 430.833 433.190

330.282 340.969 351.662 362.361 373.065

973.532 975.337 977.156 978.991 980.840

621.500 611.697 601.843 591.943 581.996

9.018 8.636 8.273 7.928 7.600

4100 4200 4300 4400 4500

107.117 107.161 107.203 107.242 107.279

529.100 531.682 534.204 536.669 539.080

435.497 437.756 439.970 442.140 444.267

383.774 394.488 405.206 415.929 426.655

982.703 1944.280 1943.998 1943.734 1943.488

572.001 547.913 514.669 481.433 448.202

7.287 6.814 6.252 5.715 5.203

4600 4700

107.313 107.345

541.438 543.746

446.354 448.402

437.384 448.117

1943.261 1943.055

414.977 381.755

4.712 4.243

580

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Boron oxide (B2O3), ideal gas, molecular weight  69.6182 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

4800 4900 5000

107.375 107.403 107.430

546.006 548.221 550.391

450.412 452.386 454.324

458.853 469.592 480.334

1942.871 1942.709 1942.570

348.538 315.323 282.113

3.793 3.361 2.947

5100 5200 5300 5400 5500

107.455 107.479 107.502 107.523 107.543

552.518 554.605 556.653 558.662 560.636

456.229 458.101 459.941 461.750 463.530

491.078 501.825 512.574 523.325 534.078

1942.457 1942.370 1942.312 1942.285 1942.289

248.904 215.700 182.494 149.291 116.088

2.549 2.167 1.799 1.444 1.103

5600 5700 5800 5900 6000

107.562 107.580 107.597 107.613 107.629

562.574 564.477 566.349 568.188 569.997

465.282 467.005 468.702 470.373 472.018

544.834 555.591 566.350 577.110 587.872

1942.328 1942.403 1942.517 1942.672 1942.870

82.883 49.678 16.470 16.736 49.949

0.773 0.455 0.148 0.148 0.435

TABLE A2 Thermochemical Data of Selected Chemical Compounds Beryllium (Be), crystal–liquid, molecular weight  9.01218 Enthalpy reference temperature  Tr  298.15 K

T (K)

Standard state pressure  p°  0.1 MPa

C°p S° [G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (J/K/mol) (J/K/mol) (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

1.932 1.893 1.315 0.723

0. 0. 0. 0.

0. 0. 0. 0.

0. 0. 0. 0.

9.440

0.

0.

0.

0.

9.440 9.649 10.136 10.792 11.551

0.030 0.909 1.872 2.899 3.975

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

13.227 14.981 16.734 18.452 20.119

6.242 8.634 11.130 13.723 16.406

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0 100 200 250

0. 1.819 9.984 13.574

0. 0.503 4.174 6.801

Infinite 19.434 10.751 9.695

298.15

16.380

9.440

300 350 400 450 500

16.472 18.520 19.965 21.061 21.943

9.542 12.245 14.817 17.235 19.501

600 700 800 900 1000

23.336 24.463 25.458 26.384 27.274

23.630 27.314 30.647 33.699 36.525

581

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Beryllium (Be), crystal–liquid, molecular weight  9.01218 Enthalpy reference temperature  Tr  298.15 K

T (K)

Standard state pressure  p°  0.1 MPa

C°p S° [G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (J/K/mol) (J/K/mol) (kJ/mol) (kJ/mol)

0. 0. 0. 0. 0.

ΔfG° (kJ/mol)

log Kf

1100 1200 1300 1400 1500

28.147 29.016 29.886 30.763 31.650

39.166 41.652 44.009 46.256 48.409

21.732 23.290 24.793 26.247 27.653

19.177 22.035 24.980 28.013 31.133

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

1527 1527

31.892 30.000

48.975 53.461

28.025 28.025

31.991 38.840

Alpha–beta transition

1560 1560

30.000 28.788

54.102 59.163

28.570 28.570

39.830 47.725

Beta–liquid transition

1600 1700 1800 1900 2000

28.874 29.089 29.304 29.519 29.734

59.893 61.650 63.319 64.909 66.429

29.344 31.193 32.932 34.573 36.128

48.879 51.777 54.696 57.638 60.600

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

2100 2200 2300 2400 2500

29.949 30.164 30.379 30.594 30.809

67.884 69.283 70.628 71.926 73.179

37.606 39.014 40.360 41.648 42.885

63.584 66.590 69.617 72.666 75.736

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

2600 2700

31.024 31.239

74.391 75.566

44.073 45.218

78.828 81.941

0. 0.

0. 0.

0. 0.

2741.437

31.328

76.043

45.680

83.237

2800 2900 3000

31.454 31.669 31.884

76.706 77.814 78.891

46.322 47.389 48.421

85.075 88.232 91.409

290.957 289.894 288.814

6.223 0.116 16.817 0.303 27.375 0.477

3100 3200 3300 3400 3500

32.099 32.314 32.529 32.744 32.959

79.940 80.963 81.960 82.934 83.887

49.421 50.391 51.333 52.248 53.138

94.608 97.829 101.071 104.335 107.620

287.718 286.609 285.488 284.356 283.215

37.897 48.383 58.834 69.251 79.635

Fugacity  1 bar

0.639 0.790 0.931 1.064 1.188

582

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Beryllium (Be), crystal–liquid, molecular weight  9.01218 Enthalpy reference temperature  Tr  298.15 K

T (K) 0 100 200 250

C°p S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol) (J/K/mol) 0. 20.786 20.786 20.786

0. 113.567 127.975 132.613

Infinite 154.755 138.176 136.617

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

6.197 4.119 2.040 1.001

319.735 321.774 323.275 323.723

319.735 310.468 298.515 292.269

Infinite 162.172 77.964 61.066

298.15

20.786

136.274

136.274

0.

324.000

286.184

50.138

300 350 400 450 500

20.786 20.786 20.786 20.786 20.786

136.403 139.607 142.383 144.831 147.021

136.275 136.528 137.090 137.817 138.630

0.038 1.078 2.117 3.156 4.196

324.008 324.169 324.245 324.257 324.221

285.950 279.592 273.218 266.839 260.460

49.788 41.727 35.679 30.974 27.210

600 700 800 900 1000

20.786 20.786 20.786 20.786 20.786

150.811 154.015 156.791 159.239 161.429

140.354 142.082 143.751 145.339 146.840

6.274 8.353 10.431 12.510 14.589

324.032 323.719 323.301 322.787 322.183

247.724 235.029 222.386 209.802 197.279

21.566 17.538 14.520 12.177 10.305

1100 1200 1300 1400 1500

20.786 20.786 20.786 20.786 20.786

163.410 165.219 166.882 168.423 169.857

148.258 149.597 150.863 152.063 153.202

16.667 18.746 20.824 22.903 24.982

321.490 320.711 319.844 318.890 317.848

184.822 172.431 160.109 147.857 135.676

8.776 7.506 6.433 5.517 4.725

1600 1700 1800 1900 2000

20.786 20.786 20.787 20.787 20.789

171.198 172.458 173.647 174.770 175.837

154.286 155.318 156.303 157.246 158.149

27.060 29.139 31.218 33.296 35.375

302.182 301.362 300.521 299.659 298.775

124.093 112.988 101.931 90.922 79.958

4.051 3.472 2.958 2.500 2.088

2100 2200 2300 2400 2500

20.791 20.795 20.801 20.811 20.824

176.851 177.818 178.743 179.628 180.478

159.016 159.849 160.650 161.423 162.168

37.454 39.533 41.613 43.694 45.775

297.870 296.943 295.996 295.028 294.039

69.040 58.165 47.332 36.541 25.791

1.717 1.381 1.075 0.795 0.539

2600 2700

20.844 20.870

181.295 182.082

162.888 163.584

47.859 49.944

293.031 292.003

15.081 4.410

0.303 0.085

2741.437

20.883

182.400

163.867

50.809

2800 2900 3000

20.905 20.949 21.006

182.842 183.576 184.287

164.259 164.912 165.546

52.033 54.126 56.223

0. 0. 0.

0. 0. 0.

0. 0. 0.

3100 3200 3300 3400 3500

21.075 21.159 21.259 21.376 21.512

184.977 185.648 186.300 186.937 187.558

166.162 166.761 167.343 167.910 168.462

58.327 60.439 62.559 64.691 66.835

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

3600 3700 3800

21.668 21.844 22.041

188.166 188.762 189.347

169.001 169.527 170.041

68.994 71.170 73.364

0. 0. 0.

0. 0. 0.

0. 0. 0.

Fugacity  1 bar

583

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Beryllium (Be), crystal–liquid, molecular weight  9.01218 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

3900 4000

22.259 22.500

189.923 190.489

170.544 171.035

75.578 77.816

0. 0.

0. 0.

0. 0.

4100 4200 4300 4400 4500

22.761 23.045 23.349 23.674 24.019

191.048 191.600 192.146 192.686 193.222

171.516 171.988 172.451 172.904 173.350

80.079 82.369 84.689 87.040 89.424

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

4600 4700 4800 4900 5000

24.383 24.764 25.162 25.576 26.003

193.754 194.282 194.808 195.331 195.852

173.788 174.218 174.642 175.058 175.469

91.844 94.301 96.798 99.334 101.913

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

5100 5200 5300 5400 5500

26.443 26.894 27.354 27.822 28.296

196.371 196.889 197.405 197.921 198.436

175.874 176.273 176.667 177.056 177.440

104.535 107.202 109.914 112.673 115.479

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

5600 5700 5800 5900 6000

28.776 29.258 29.743 30.227 30.711

198.950 199.464 199.977 200.489 201.001

177.819 178.195 178.566 178.933 179.296

118.333 121.234 124.184 127.183 130.230

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

TABLE A2 Thermochemical Data of Selected Chemical Compounds Beryllium oxide (BeO), ideal gas, molecular weight  25.01158 Enthalpy reference temperature  Tr  298.15 K

T (K) 0 100 200

C°p S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol) (J/K/mol) 0. 2.636 14.159

0. 0.824 5.887

Infinite 28.539 15.759

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol) 2.835 2.771 1.974

604.914 606.343 607.579

ΔfG° (kJ/mol) 604.914 597.709 588.573

log Kf Infinite 312.211 153.719

298.15

25.560

13.770

13.770

0.

608.354

579.062

101.449

300 400 500

25.744 33.757 38.920

13.928 22.513 30.641

13.770 14.886 17.235

0.047 3.051 6.703

608.364 608.688 608.668

578.880 568.992 559.065

100.792 74.303 58.405

600 700

42.376 44.823

38.061 44.786

20.098 23.152

10.778 15.144

608.440 608.092

549.163 539.310

47.809 40.244

584

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Beryllium oxide (BeO), ideal gas, molecular weight  25.01158 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

800 900 1000

46.656 48.091 49.262

50.897 56.478 61.607

26.244 29.298 32.276

19.722 24.462 29.331

607.680 607.235 606.780

529.511 519.767 510.073

34.574 30.166 26.643

1100 1200 1300 1400 1500

50.254 51.116 51.882 52.580 53.225

66.350 70.760 74.882 78.753 82.403

35.161 37.946 40.630 43.216 45.708

34.308 39.377 44.527 49.751 55.042

606.329 605.892 605.478 605.094 604.744

500.424 490.815 481.243 471.701 462.185

23.763 21.365 19.337 17.599 16.095

1600 1700 1800 1900 2000

53.827 54.396 54.944 55.467 55.974

85.857 89.138 92.262 95.247 98.105

48.111 50.428 52.666 54.829 56.922

60.394 65.806 71.273 76.794 82.366

618.971 618.303 617.614 616.904 616.176

452.166 441.761 431.396 421.069 410.781

14.762 13.574 12.519 11.576 10.729

2100 2200 2300

56.467 56.948 57.421

100.848 103.486 106.028

58.949 60.914 62.820

87.988 93.659 99.378

615.430 614.669 613.893

400.530 390.314 380.133

9.963 9.267 8.633

2373.001 2373.001

57.756 57.758

107.827 110.649

64.177 64.177

103.582 110.276

2400 2500

57.881 58.338

111.303 113.675

64.704 66.616

111.837 117.648

606.409 605.605

370.063 360.232

8.054 7.527

2600 2700 2800

58.789 59.233 59.676

115.972 118.199 120.361

68.470 70.271 72.021

123.504 129.406 135.351

604.789 603.959 894.074

350.433 340.665 324.707

7.040 6.591 6.057

2821.220 2821.220

59.770 79.496

120.812 148.834

72.386 72.386

136.618 215.676

2900 3000

79.496 79.496

151.024 153.719

74.493 77.089

221.939 229.888

811.558 807.695

306.635 289.290

5.523 5.037

3100 3200 3300 3400 3500

79.496 79.496 79.496 79.496 79.496

156.325 158.849 161.295 163.669 165.973

79.603 82.041 84.405 86.702 88.934

237.838 245.787 253.737 261.687 269.636

803.847 800.016 796.202 792.408 788.635

272.073 254.980 238.007 221.149 204.402

4.584 4.162 3.767 3.398 3.051

3600 3700 3800 3900 4000

79.496 79.496 79.496 79.496 79.496

168.213 170.391 172.511 174.576 176.588

91.105 93.219 95.278 97.285 99.242

277.586 285.535 293.485 301.435 309.384

784.884 781.157 777.455 773.782 770.138

187.763 171.228 154.792 138.454 122.211

2.724 2.417 2.128 1.854 1.596

4100 4200 4300 4400 4500

79.496 79.496 79.496 79.496 79.496

178.551 180.467 182.337 184.165 185.952

101.153 103.018 104.841 106.624 108.367

317.334 325.283 333.233 341.183 349.132

766.525 762.946 759.403 755.897 752.431

106.057 89.992 74.011 58.112 42.293

1.351 1.119 0.899 0.690 0.491

Alpha–beta transition

Beta–liquid transition

585

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Beryllium oxide (BeO), ideal gas, molecular weight  25.01158 Enthalpy reference temperature  Tr  298.15 K

T (K) 4600 4700 4800 4900 5000

C°p S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol) (J/K/mol) 79.496 79.496 79.496 79.496 79.496

187.699 189.408 191.082 192.721 194.327

110.072 111.742 113.378 114.980 116.551

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol) 357.082 365.031 372.981 380.931 388.880

749.007 745.626 742.290 739.001 735.761

ΔfG° (kJ/mol) 26.550 10.881 4.717 20.246 35.708

log Kf 0.301 0.121 0.051 0.216 0.373

TABLE A2 Thermochemical Data of Selected Chemical Compounds Beryllium oxide (BeO), ideal gas, molecular weight  25.01158 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p (J/K/mol)

S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

log Kf

8.688 5.784 2.873

133.984 135.397 136.275

133.984 127.536 119.270

Infinite 66.618 31.150

197.625

0.

136.398

110.873

19.425

197.626 198.791 201.023

0.055 3.043 6.131

136.395 136.056 135.513

110.715 102.198 93.793

19.277 13.346 9.799

219.108 224.179 228.682 232.733 236.418

203.564 206.155 208.694 211.144 213.490

9.326 12.617 15.990 19.430 22.928

134.860 134.133 133.340 132.485 131.569

85.509 77.340 69.280 61.324 53.465

7.444 5.771 4.524 3.559 2.793

35.784 36.357 37.023 37.825 38.788

239.803 242.941 245.876 248.648 251.290

215.730 217.869 219.911 221.866 223.740

26.480 30.086 33.755 37.496 41.325

130.595 129.569 128.501 127.403 126.291

45.702 38.029 30.444 22.942 15.519

2.170 1.655 1.223 0.856 0.540

1600 1700 1800 1900 2000

39.927 41.236 42.698 44.285 45.956

253.829 256.288 258.685 261.036 263.350

225.542 227.278 228.957 230.584 232.164

45.259 49.316 53.511 57.860 62.371

110.646 109.959 109.377 108.914 108.582

8.696 2.346 3.967 10.250 16.513

0.284 0.072 0.115 0282 0.431

2100 2200 2300 2400 2500

47.670 49.379 51.042 52.616 54.070

265.633 267.890 270.122 272.328 274.506

233.704 235.206 236.676 238.116 239.528

67.052 71.905 76.927 82.110 87.446

108.386 108.329 108.408 108.617 108.944

22.762 29.005 35.249 41.499 47.761

0.566 0.689 0.801 0.903 0.998

0 100 200

0. 29.108 29.139

0. 165.763 185.943

Infinite 223.605 200.307

298.15

29.481

197.625

300 400 500

29.493 30.348 31.421

197.807 206.400 213.286

600 700 800 900 1000

32.454 33.345 34.084 34.704 35.252

1100 1200 1300 1400 1500

586

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Beryllium oxide (BeO), ideal gas, molecular weight  25.01158 Enthalpy reference temperature  Tr  298.15 K S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

T (K)

C°p (J/K/mol)

2600 2700 2800 2900 3000

55.375 56.514 57.474 58.255 58.856

276.653 278.765 280.838 282.869 284.854

240.915 242.277 243.618 244.936 246.234

92.919 98.515 104.216 110.004 115.861

109.378 109.902 180.457 178.741 176.970

54.037 60.332 60.425 56.168 51.972

1.086 1.167 1.127 1.012 0.905

3100 3200 3300 3400 3500

59.286 59.556 59.681 59.676 59.553

286.792 288.679 290.514 292.295 294.024

247.511 248.768 250.006 251.223 252.422

121.770 127.713 133.676 139.645 145.608

175.163 173.338 171.511 169.698 167.911

47.834 43.757 39.735 35.769 31.856

0.806 0.714 0.629 0.550 0.475

3600 3700 3800 3900 4000

59.345 59.051 58.691 58.279 57.827

295.699 297.321 298.891 300.410 301.880

253.601 254.760 255.901 257.023 258.126

151.554 157.474 163.362 169.210 175.016

166.164 164.466 162.827 161.254 159.754

27.994 24.180 20.410 16.682 12.995

0.406 0.341 0.281 0.223 0.170

4100 4200 4300 4400 4500

57.347 56.848 56.337 55.822 55.308

303.302 304.678 306.010 307.299 308.548

259.211 260.277 261.325 262.356 263.368

180.775 186.485 192.144 197.752 203.309

158.332 156.993 155.740 154.576 153.503

9.344 5.726 2.139 1.419 4.952

0.119 0.071 0.026 0.017 0.057

4600 4700 4800 4900 5000

54.801 54.303 53.818 53.348 52.895

309.758 310.931 312.069 313.174 314.247

264.364 265.342 266.304 267.249 268.178

208.814 214.269 219.675 225.033 230.345

152.523 151.636 150.844 150.146 149.544

8.462 11.952 15.424 18.881 22.324

0.096 0.133 0.168 0.201 0.233

5100 5200 5300 5400 5500

52.460 52.044 51.647 51.271 50.914

315.291 316.305 317.293 318.255 319.192

269.092 269.990 270.873 271.742 272.596

235.613 240.838 246.022 251.168 256.277

149.035 148.621 148.301 148.073 147.937

25.756 29.179 32.595 36.006 39.413

0.264 0.293 0.321 0.348 0.374

5600 5700 5800 5900 6000

50.577 50.259 49.961 49.681 49.418

320.106 320.999 321.870 322.722 323.555

273.436 274.263 275.076 275.877 276.665

261.351 266.393 271.404 276.386 281.340

147.892 147.936 148.069 148.289 148.596

42.819 46.225 49.633 53.042 56.458

0.399 0.424 0.447 0.470 0.492

log Kf

587

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Carbon (C), reference state-graphite, molecular weight  12.011 Enthalpy reference temperature  Tr  298.15 K [G°  H°(Tr)]/T (J/K/mol)

C°p (J/K/mol)

S° (J/K/mol)

0 100 200 250

0. 1.674 5.006 6.816

0. 0.952 3.082 4.394

Infinite 10.867 6.407 5.871

298.15

8.517

5.740

300 350 400 450 500

8.581 10.241 11.817 13.289 14.623

5.793 7.242 8.713 10.191 11.662

600 700 800 900 1000

16.844 18.537 19.827 20.824 21.610

1100 1200 1300 1400 1500

T (K)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

log Kf

1.051 0.991 0.665 0.369

0. 0. 0. 0.

0. 0. 0. 0.

0. 0. 0. 0.

5.740

0.

0.

0.

0.

5.741 5.851 6.117 6.487 6.932

0.016 0.487 1.039 1.667 2.365

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

14.533 17.263 19.826 22.221 24.457

7.961 9.097 10.279 11.475 12.662

3.943 5.716 7.637 9.672 11.795

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

22.244 22.766 23.204 23.578 23.904

26.548 28.506 30.346 32.080 33.718

13.831 14.973 16.085 17.167 18.216

13.989 16.240 18.539 20.879 23.253

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

1600 1700 1800 1900 2000

24.191 24.448 24.681 24.895 25.094

35.270 36.744 38.149 39.489 40.771

19.234 20.221 21.178 22.107 23.008

25.658 28.090 30.547 33.026 35.525

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

2100 2200 2300 2400 2500

25.278 25.453 25.618 25.775 25.926

42.000 43.180 44.315 45.408 46.464

23.883 24.734 25.561 26.365 27.148

38.044 40.581 43.134 45.704 48.289

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

2600 2700 2800 2900 3000

26.071 26.212 26.348 26.481 26.611

47.483 48.470 49.426 50.353 51.253

27.911 28.654 29.379 30.086 30.777

50.889 53.503 56.131 58.773 61.427

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

3100 3200 3300 3400 3500

26.738 26.863 26.986 27.106 27.225

52.127 52.978 53.807 54.614 55.401

31.451 32.111 32.756 33.387 34.005

64.095 66.775 69.467 72.172 74.889

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

3600 3700 3800 3900 4000

27.342 27.459 27.574 27.688 27.801

56.170 56.921 57.655 58.372 59.075

34.610 35.203 35.784 36.354 38.913

77.617 80.357 83.109 85.872 88.646

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

588

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Carbon (C), reference state-graphite, molecular weight  12.011 Enthalpy reference temperature  Tr  298.15 K [G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

T (K)

C°p (J/K/mol)

S° (J/K/mol)

4100 4200 4300 4400 4500

27.913 28.024 28.134 28.245 28.354

59.763 60.437 61.097 61.745 62.381

37.462 38.001 38.531 39.051 39.562

91.432 94.229 97.037 99.856 102.685

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

4600 4700 4800 4900 5000

28.462 28.570 28.678 28.785 28.893

63.006 63.619 64.222 64.814 65.397

40.065 40.560 41.047 41.526 41.997

105.526 108.378 111.240 114.114 116.997

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

5100 5200 5300 5400 5500

28.999 29.106 29.211 29.317 29.422

65.970 66.534 67.089 67.636 68.175

42.462 42.919 43.370 43.814 44.252

119.892 122.797 125.713 128.640 131.577

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

5600 5700 5800 5900 6000

29.528 29.632 29.737 29.842 29.946

68.706 69.230 69.746 70.255 70.758

44.684 45.110 45.531 45.945 46.355

134.524 137.482 140.451 143.429 146.419

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

log Kf

TABLE A2 Thermochemical Data of Selected Chemical Compounds Carbon (C), ideal gas, molecular weight  12.011 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

6.536 4.150 2.048 1.004

711.185 713.511 715.287 716.035

711.185 700.088 685.950 678.527

Infinite 365.689 179.152 141.770

158.100

0.

716.670

671.244

117.599

158.100 158.354 158.917 159.645 160.459

0.039 1.080 2.121 3.162 4.202

716.693 717.263 717.752 718.165 718.507

670.962 663.294 655.550 647.749 639.906

116.825 98.991 85.606 75.189 66.851

0 100 200 250

0. 21.271 20.904 20.861

0. 135.180 149.768 154.427

Infinite 176.684 160.007 158.443

298.15

20.838

158.100

300 350 400 450 500

20.838 20.824 20.815 20.809 20.804

158.228 161.439 164.219 166.671 168.863

589

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Carbon (C), ideal gas, molecular weight  12.011 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

600 700 800 900 1000

20.799 20.795 20.793 20.792 20.791

172.655 175.861 178.638 181.087 183.278

162.185 163.916 165.587 167.175 168.678

6.282 8.362 10.441 12.520 14.600

719.009 719.315 719.474 719.519 719.475

624.135 608.296 592.424 576.539 560.654

54.336 45.392 38.681 33.461 29.286

1100 1200 1300 1400 1500

20.791 20.793 20.796 20.803 20.814

185.259 187.068 188.733 190.274 191.710

170.097 171.437 172.704 173.905 175.044

16.679 18.758 20.837 22.917 24.998

719.360 719.188 718.968 718.709 718.415

544.777 528.913 513.066 497.237 481.427

25.869 23.023 20.615 18.552 16.765

1600 1700 1800 1900 2000

20.829 20.850 20.878 20.912 20.952

193.053 194.317 195.509 196.639 197.713

176.128 177.162 178.148 179.092 179.996

27.080 29.164 31.250 33.340 35.433

718.092 717.744 717.373 716.984 716.577

465.639 449.871 434.124 418.399 402.694

15.202 13.823 12.598 11.503 10.517

2100 2200 2300 2400 2500

20.999 21.052 21.110 21.174 21.241

198.736 199.714 200.651 201.551 202.417

180.864 181.699 182.503 183.278 184.026

37.530 39.633 41.741 43.855 45.976

716.156 715.722 715.277 714.821 714.357

387.010 371.347 355.703 340.079 324.474

9.626 8.817 8.078 7.402 6.780

2600 2700 2800 2900 3000

21.313 21.387 21.464 21.542 21.621

203.251 204.057 204.836 205.591 206.322

184.750 185.450 186.129 186.787 187.426

48.103 50.238 52.381 54.531 56.689

713.884 713.405 712.920 712.429 711.932

308.888 293.321 277.771 262.239 246.723

6.206 5.675 5.182 4.723 4.296

3100 3200 3300 3400 3500

21.701 21.780 21.859 21.936 22.012

207.032 207.723 208.394 209.048 209.685

188.047 188.651 189.239 189.812 190.371

58.856 61.030 63.212 65.401 67.599

711.431 710.925 710.414 709.899 709.380

231.224 215.742 200.275 184.824 169.389

3.896 3.522 3.170 2.839 2.528

3600 3700 3800 3900 4000

22.087 22.159 22.230 22.298 22.363

210.306 210.912 211.504 212.082 212.648

190.916 191.448 191.968 192.477 192.974

69.804 72.016 74.235 76.462 78.695

708.857 708.329 707.797 707.260 706.719

153.968 138.561 123.169 107.791 92.427

2.234 1.956 1.693 1.444 1.207

4100 4200 4300 4400 4500

22.426 22.487 22.544 22.600 22.652

213.201 213.742 214.272 214.791 215.299

193.461 193.937 194.404 194.861 195.310

80.934 83.180 85.432 87.689 89.951

706.173 705.621 705.065 704.503 703.936

77.077 61.740 46.416 31.105 15.807

0.982 0.768 0.564 0.369 0.183

4600 4700 4800 4900 5000

22.702 22.750 22.795 22.838 22.878

215.797 216.286 216.766 217.236 217.698

195.750 196.182 196.605 197.022 197.431

92.219 94.492 96.769 99.051 101.337

703.363 702.784 702.199 701.607 701.009

0.521 14.752 30.012 45.261 60.497

0.006 0.164 0.327 0.482 0.632

590

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Carbon (C), ideal gas, molecular weight  12.011 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

5100 5200 5300 5400 5500

22.917 22.953 22.987 23.020 23.051

218.151 218.597 219.034 219.464 219.887

197.832 198.227 198.616 196.998 199.374

103.626 105.920 108.217 110.517 112.821

700.404 699.793 699.174 698.548 697.914

75.721 90.933 106.133 121.322 136.499

0.776 0.913 1.046 1.174 1.296

5600 5700 5800 5900 6000

23.080 23.107 23.133 23.157 23.181

220.302 220.711 221.113 221.509 221.898

199.744 200.108 200.467 200.820 201.168

115.127 117.437 119.749 122.063 124.380

697.273 696.625 695.968 695.304 694.631

151.664 166.818 181.961 197.092 212.211

1.415 1.529 1.639 1.745 1.847

TABLE A2 Thermochemical Data of Selected Chemical Compounds Methane (CH4), ideal gas, molecular weight  16.04276 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) (kJ/mol) 10.024 6.698 3.368 1.679

ΔfH° (kJ/mol) 66.911 69.644 72.027 73.426

ΔfG° (kJ/mol) 66.911 64.353 58.161 54.536

log Kf

0 100 200 250

0. 33.258 33.473 34.216

0. 149.500 172.577 180.113

Infinite 216.485 189.418 186.829

Infinite 33.615 15.190 11.395

298.15

35.639

186.251

186.251

0.

74.873

50.768

8.894

300 350 400 450 500

35.708 37.874 40.500 43.374 46.342

186.472 192.131 197.356 202.291 207.014

186.252 186.694 187.704 189.053 190.614

0.066 1.903 3.861 5.957 8.200

74.929 76.461 77.969 79.422 80.802

50.618 46.445 42.054 37.476 32.741

8.813 6.932 5.492 4.350 3.420

600 700 800 900 1000

52.227 57.794 62.932 67.601 71.795

215.987 224.461 232.518 240.205 247.549

194.103 197.840 201.675 205.532 209.370

13.130 18.635 24.675 31.205 38.179

83.308 85.452 87.238 88.692 89.849

22.887 12.643 2.115 8.616 19.492

1.993 0.943 0.138 0.500 1.018

1100 1200 1300 1400 1500

75.529 78.833 81.744 84.305 86.556

254.570 261.287 267.714 273.868 279.763

213.162 216.895 220.558 224.148 227.660

45.549 53.270 61.302 69.608 78.153

90.750 91.437 91.945 92.308 92.553

30.472 41.524 52.626 63.761 74.918

1.447 1.807 2.115 2.379 2.609

591

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Methane (CH4), ideal gas, molecular weight  16.04276 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

log Kf

1600 1700 1800 1900 2000

88.537 90.283 91.824 93.188 94.399

285.413 290.834 296.039 301.041 305.853

231.095 234.450 237.728 240.930 244.057

86.910 95.853 104.960 114.212 123.592

92.703 92.780 92.797 92.770 92.709

86.088 97.265 108.445 119.624 130.802

2.810 2.989 3.147 3.289 3.416

2100 2200 2300 2400 2500

95.477 96.439 97.301 98.075 98.772

310.485 314.949 319.255 323.413 327.431

247.110 250.093 253.007 255.854 258.638

133.087 142.684 152.371 162.141 171.984

92.624 92.521 92.409 92.291 92.174

141.975 153.144 164.308 175.467 186.622

3.531 3.636 3.732 3.819 3.899

2600 2700 2800 2900 3000

99.401 99.971 100.489 100.960 101.389

331.317 335.080 338.725 342.260 345.690

261.359 264.020 266.623 269.171 271.664

181.893 191.862 201.885 211.958 222.076

92.060 91.954 91.857 91.773 91.705

197.771 208.916 220.058 231.196 242.332

3.973 4.042 4.105 4.164 4.219

3100 3200 3300 3400 3500

101.782 102.143 102.474 102.778 103.060

349.021 352.258 355.406 358.470 361.453

274.106 276.498 278.842 281.139 283.391

232.235 242.431 252.662 262.925 273.217

91.653 91.621 91.609 91.619 91.654

253.465 264.598 275.730 286.861 297.993

4.271 4.319 4.364 4.407 4.447

3600 3700 3800 3900 4000

103.319 103.560 103.783 103.990 104.183

364.360 367.194 369.959 372.658 375.293

285.600 287.767 289.894 291.982 294.032

283.536 293.881 304.248 314.637 325.045

91.713 91.798 91.911 92.051 92.222

309.127 320.262 331.401 342.542 353.687

4.485 4.521 4.555 4.588 4.619

4100 4200 4300 4400 4500

104.363 104.531 104.688 104.834 104.972

377.868 380.385 382.846 385.255 387.612

296.045 298.023 299.967 301.879 303.758

335.473 345.918 356.379 366.855 377.345

92.422 92.652 92.914 93.208 93.533

364.838 375.993 387.155 398.322 409.497

4.648 4.676 4.703 4.729 4.753

4600 4700 4800 4900 5000

105.101 105.223 105.337 105.445 105.546

389.921 392.182 394.399 396.572 398.703

305.606 307.424 309.213 310.973 312.707

387.849 398.365 408.893 419.432 429.982

93.891 94.281 94.702 95.156 95.641

420.679 431.869 443.069 454.277 465.495

4.777 4.800 4.822 4.843 4.863

5100 5200 5300 5400 5500

105.642 105.733 105.818 105.899 105.976

400.794 402.847 404.861 406.840 408.784

314.414 316.095 317.750 319.382 320.990

440.541 451.110 461.688 472.274 482.867

96.157 96.703 97.278 97.882 98.513

476.722 487.961 499.210 510.470 521.741

4.883 4.902 4.920 4.938 4.955

5600 5700 5800 5900 6000

106.049 106.118 106.184 106.247 106.306

410.694 412.572 414.418 416.234 418.020

322.575 324.137 325.678 327.197 328.696

493.469 504.077 514.692 525.314 535.942

99.170 99.852 100.557 101.284 102.032

533.025 544.320 555.628 566.946 578.279

4.972 4.988 5.004 5.019 5.034

592

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Carbon monoxide (CO), ideal gas, molecular weight  28.0104 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

0 100 200

0. 29.104 29.108

0. 165.850 186.025

Infinite 223.539 200.317

298.15

29.142

197.653

197.653

0.

110.527

137.163

24.030

300 400 500

29.142 29.342 29.794

197.833 206.238 212.831

197.653 198.798 200.968

0.054 2.976 5.931

110.516 110.102 110.003

137.328 146.338 155.414

23.911 19.110 16.236

600 700 800 900 1000

30.443 31.171 31.899 32.577 33.183

218.319 223.066 227.277 231.074 234.538

203.415 205.890 208.305 210.628 212.848

8.942 12.023 15.177 18.401 21.690

110.150 110.469 110.905 111.418 111.983

164.486 173.518 182.497 191.416 200.275

14.320 12.948 11.916 11.109 10.461

1100 1200 1300 1400 1500

33.710 34.175 34.572 34.920 35.217

237.726 240.679 243.431 246.006 248.426

214.967 216.988 218.917 220.761 222.526

25.035 28.430 31.868 35.343 38.850

112.586 113.217 113.870 114.541 115.229

209.075 217.819 226.509 235.149 243.740

9.928 9.481 9.101 8.774 8.488

1600 1700 1800 1900 2000

35.480 35.710 35.911 36.091 36.250

250.707 252.865 254.912 256.859 258.714

224.216 225.839 227.398 228.897 230.342

42.385 45.945 49.526 53.126 56.744

115.933 116.651 117.384 118.133 118.896

252.284 260.784 269.242 277.658 286.034

8.236 8.013 7.813 7.633 7.470

2100 2200 2300 2400 2500

36.392 36.518 36.635 36.321 36.836

260.486 262.182 263.809 265.359 266.854

231.736 233.081 234.382 235.641 236.860

60.376 64.021 67.683 71.324 74.985

119.675 120.470 121.278 122.133 122.994

294.372 302.672 310.936 319.164 327.356

7.322 7.186 7.062 6.946 6.840

2600 2700 2800 2900 3000

36.924 37.003 37.083 37.150 37.217

268.300 269.695 271.042 272.345 273.605

238.041 239.188 240.302 241.384 242.437

78.673 82.369 86.074 89.786 93.504

123.854 124.731 125.623 126.532 127.457

335.514 343.638 351.729 359.789 367.816

6.741 6.648 6.562 6.480 6.404

3100 3200 3300 3400 3500

37.279 37.338 37.392 37.443 37.493

274.827 276.011 277.161 278.278 279.364

243.463 244.461 245.435 246.385 247.311

97.229 100.960 104.696 108.438 112.185

128.397 129.353 130.325 131.312 132.313

375.812 383.778 391.714 399.620 407.497

6.332 6.265 6.200 6.139 6.082

3600 3700 3800 3900 4000

37.543 37.589 37.631 37.673 37.715

280.421 281.450 282.453 283.431 284.386

248.216 249.101 249.965 250.811 251.638

115.937 119.693 123.454 127.219 130.989

133.329 134.360 135.405 136.464 137.537

415.345 423.165 430.956 438.720 446.457

6.027 5.974 5.924 5.876 5.830

T (K)

H°  H°(Tr) (kJ/mol) 8.671 5.769 2.858

ΔfH° (kJ/mol) 113.805 112.415 111.286

ΔfG° (kJ/mol)

log Kf

113.805 Infinite 120.239 62.807 128.526 33.568

593

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Carbon monoxide (CO), ideal gas, molecular weight  28.0104 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

4100 4200 4300 4400 4500

37.756 37.794 37.832 37.869 37.903

285.317 286.228 287.117 287.988 288.839

252.449 253.242 254.020 254.782 255.529

134.762 138.540 142.321 146.106 149.895

138.623 139.723 140.836 141.963 143.103

454.166 461.849 469.506 477.136 484.741

5.786 5.744 5.703 5.664 5.627

4600 4700 4800 4900 5000

37.941 37.974 38.007 38.041 38.074

289.673 290.489 291.289 292.073 292.842

256.262 256.982 257.688 258.382 259.064

153.687 157.483 161.282 165.084 168.890

144.257 145.424 146.605 147.800 149.009

492.321 499.875 507.404 514.908 522.387

5.590 5.555 5.522 5.489 5.457

5100 5200 5300 5400 5500

38.104 38.137 38.171 38.200 38.074

293.596 294.336 295.063 295.777 296.476

259.733 260.392 261.039 261.676 262.302

172.699 176.511 180.326 184.146 187.957

150.231 151.469 152.721 153.987 155.279

529.843 537.275 544.681 552.065 559.426

5.427 5.397 5.368 5.340 5.313

5600 5700 5800 5900 6000

38.263 38.296 38.325 38.355 38.388

297.164 297.842 298.508 299.163 299.808

262.919 263.525 264.123 264.711 265.291

191.775 195.603 199.434 203.268 207.106

156.585 157.899 159.230 160.579 161.945

566.762 574.075 581.364 588.631 595.875

5.287 5.261 5.236 5.211 5.188

log Kf

TABLE A2 Thermochemical Data of Selected Chemical Compounds Carbon dioxide (CO2), ideal gas, molecular weight  44.0098 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p (J/K/mol)

S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

0 100 200

0. 29.20 8 32.359

0. 179.009 199.975

Infinite 243.568 217.046

9.364 6.456 3.414

393.151 393.208 393.404

393.151 393.683 394.085

Infinite 205.639 102.924

298.15

37.129

213.795

213.795

0.

393.522

394.389

69.095

300 400 500

37.221 41.325 44.627

214.025 225.314 234.901

213.795 215.307 218.290

0.069 4.003 8.305

393.523 393.583 393.666

394.394 394.675 394.939

68.670 51.539 41.259

600 700 800 900 1000

47.321 49.564 51.434 52.999 54.308

243.283 250.750 257.494 263.645 269.299

221.772 225.388 228.986 232.500 235.901

12.907 17.754 22.806 28.030 33.397

393.803 393.983 394.168 394.405 394.623

395.182 395.398 395.586 395.748 395.886

34.404 29.505 25.829 22.969 20.679

594

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Carbon dioxide (CO2), ideal gas, molecular weight  44.0098 Enthalpy reference temperature  Tr  298.15 K S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

T (K)

C°p (J/K/mol)

1100 1200 1300 1400 1500

55.409 56.342 57.137 57.802 58.379

274.528 279.390 283.932 288.191 292.199

239.178 242.329 245.356 248.265 251.062

38.884 44.473 50.148 55.896 61.705

394.838 395.050 395.257 395.462 395.668

396.001 396.098 396.177 396.240 396.288

18.805 17.242 15.919 14.784 13.800

1600 1700 1800 1900 2000

58.886 59.317 59.701 60.049 60.350

295.983 299.566 302.968 306.205 309.293

253.753 256.343 258.840 261.248 263.574

67.569 73.480 79.431 85.419 91.439

395.876 396.090 396.311 396.542 396.784

396.323 396.344 396.353 396.349 396.333

12.939 12.178 11.502 10.896 10.351

2100 2200 2300 2400 2500

60.622 60.865 61.086 61.287 61.471

312.244 315.070 317.781 320.385 322.890

265.822 267.996 270.102 272.144 274.124

97.488 103.562 109.660 115.779 121.917

397.039 397.309 397.596 397.900 398.222

396.304 396.262 396.209 396.142 396.062

9.858 9.408 8.998 8.622 8.275

2600 2700 2800 2900 3000

61.647 61.802 61.952 62.095 62.229

325.305 327.634 329.885 332.061 334.169

276.046 277.914 279.730 281.497 283.218

128.073 134.246 140.433 146.636 152.852

398.562 398.921 399.299 399.695 400.111

395.969 395.862 395.742 395.609 395.461

7.955 7.658 7.383 7.126 6.886

3100 3200 3300 3400 3500

62.347 62.462 62.573 62.681 62.785

336.211 338.192 340.116 341.986 343.804

284.895 286.529 288.124 289.681 291.202

159.081 165.321 171.573 177.836 184.109

400.545 400.998 401.470 401.960 402.467

395.298 395.122 394.932 394.726 394.506

6.661 6.450 6.251 6.064 5.888

3600 3700 3800 3900 4000

62.884 62.980 63.074 63.166 63.254

345.574 347.299 348.979 350.619 352.219

292.687 294.140 295.561 296.952 298.314

190.393 196.686 202.989 209.301 215.622

402.991 403.532 404.089 404.662 405.251

394.271 394.022 393.756 393.477 393.183

5.721 5.563 5.413 5.270 5.134

4100 4200 4300 4400 4500

63.341 63.426 63.509 63.588 63.667

353.782 355.310 356.803 358.264 359.694

299.648 300.955 302.236 303.493 304.726

221.951 228.290 234.637 240.991 247.354

405.856 406.475 407.110 407.760 408.426

392.874 392.550 392.210 391.857 391.488

5.005 4.882 4.764 4.652 4.544

4600 4700 4800 4900 5000

63.745 63.823 63.893 63.968 64.046

361.094 362.466 363.810 365.126 366.422

305.937 307.125 308.292 309.438 310.565

253.725 260.103 266.489 272.882 279.283

409.106 409.802 410.514 411.242 411.986

391.105 390.706 390.292 389.892 389.419

4.441 4.342 4.247 4.156 4.068

5100 5200 5300 5400 5500

64.128 64.220 64.312 64.404 64.496

367.691 368.937 370.161 371.364 372.547

311.673 312.762 313.833 314.888 315.925

285.691 292.109 298.535 304.971 311.416

412.746 413.522 414.314 415.123 415.949

388.959 388.486 387.996 387.493 386.974

3.984 3.902 3.824 3.748 3.675

log Kf

595

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Carbon dioxide (CO2), ideal gas, molecular weight  44.0098 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p (J/K/mol)

5600 5700 5800 5900 6000

64.588 64.680 64.772 64.865 64.957

S° [G°  H°(Tr)]/T (J/K/mol) (J/K/mol) 373.709 374.853 375.979 377.087 378.178

316.947 317.953 318.944 319.920 320.882

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol) 317.870 324.334 330.806 337.288 343.779

416.794 417.658 418.541 419.445 420.372

ΔfG° (kJ/mol) 386.439 385.890 385.324 384.745 384.148

log Kf 3.605 3.536 3.470 3.406 3.344

TABLE A2 Thermochemical Data of Selected Chemical Compounds Acetylene (C2H2), ideal gas, molecular weight  26.03788 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

10.012 7.104 3.925

235.755 232.546 229.685

235.755 236.552 241.663

Infinite 123.562 63.116

log Kf

0 100 200

0. 29.347 35.585

0. 163.294 185.097

Infinite 234.338 204.720

298.15

44.095

200.958

200.958

0.

226.731

248.163

43.477

300 400 500

44.229 50.480 54.869

201.231 214.856 226.610

200.959 202.774 206.393

0.082 4.833 10.108

226.674 223.568 220.345

248.296 255.969 264.439

43.232 33.426 27.626

600 700 800 900 1000

58.287 61.149 63.760 66.111 68.275

236.924 246.127 254.466 262.113 269.192

210.640 215.064 219.476 223.794 227.984

15.771 21.745 27.992 34.487 41.208

216.993 213.545 210.046 206.522 202.989

273.571 283.272 293.471 304.111 315.144

23.816 21.138 19.162 17.650 16.461

1100 1200 1300 1400 1500

70.245 72.053 73.693 75.178 76.530

275.793 281.984 287.817 293.334 298.567

232.034 235.941 239.709 243.344 246.853

48.136 55.252 62.540 69.985 77.572

199.451 195.908 192.357 188.795 185.216

326.530 338.239 350.244 362.523 375.057

15.506 14.723 14.073 13.526 13.061

1600 1700 1800 1900 2000

77.747 78.847 79.852 80.760 81.605

303.546 308.293 312.829 317.171 321.335

250.242 253.518 256.688 259.758 262.733

85.286 93.117 101.053 109.084 117.203

181.619 177.998 174.353 170.680 166.980

387.830 400.829 414.041 427.457 441.068

12.661 12.316 12.015 11.752 11.520

2100 2200 2300 2400 2500

82.362 83.065 83.712 84.312 84.858

325.335 329.183 332.890 336.465 339.918

265.620 268.422 271.145 273.793 276.369

125.401 133.673 142.012 150.414 158.873

163.250 159.491 155.701 151.881 148.029

454.864 468.838 482.984 497.295 511.767

11.314 11.132 10.969 10.823 10.693

596

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Acetylene (C2H2), ideal gas, molecular weight  26.03788 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

log Kf

2600 2700 2800 2900 3000

85.370 85.846 86.295 86.713 87.111

343.256 346.487 349.618 352.653 355.600

278.878 281.322 283 706 286 031 288.301

167.384 175.945 184.553 193.203 201.895

144.146 140.230 136.282 132.302 128.290

526.393 541.169 556.090 571.154 586.355

10.575 10.470 10.374 10.288 10.209

3100 3200 3300 3400 3500

87.474 87.825 88.164 88.491 88.805

358.462 361.245 363.952 366.589 369.159

290.519 292.686 294.804 296.877 298.906

210.624 219.389 228.189 237.022 245.886

124.245 120.166 116.053 111.908 107.731

601.690 617.157 632.751 648.471 664.313

10.138 10.074 10.016 9.963 9.914

3600 3700 3800 3900 4000

89.101 89.388 89.666 89.935 90.194

371.665 374.110 376.498 378.830 381.110

300.892 302.838 304.745 306.615 308.449

254.782 263.706 272.659 281.639 290.646

103.519 99.274 94.996 90.683 86.336

680.275 696.354 712.549 728.856 745.275

9.871 9.831 9.795 9.762 9.732

4100 4200 4300 4400 4500

90.439 90.678 90.910 91.137 91.358

383.341 385.523 387.659 389.752 391.802

310.248 312.015 313.749 315.453 317.127

299.678 308.733 317.813 326.915 336.040

81.955 77.538 73.087 68.601 64.080

761.803 778.438 795.178 812.023 828.969

9.705 9.681 9.660 9.640 9.622

4600 4700 4800 4900 5000

91.563 91.768 91.970 92.171 92.370

393.813 395.784 397.718 399.617 401.481

318.772 320.390 321.981 323.546 325.086

345.186 354.353 363.540 372.747 381.974

59.524 54.933 50.307 45.648 40.957

846.017 863.164 880.410 897.751 915.189

9.607 9.593 9.581 9.570 9.561

5100 5200 5300 5400 5500

92.571 92.768 92.963 93.153 93.341

403.312 405.111 406.880 408.620 410.331

326.602 328.094 329.564 331.012 332.439

391.221 400.488 409.774 419.080 428.405

36.234 31.481 26.699 21.889 17.052

932.720 950.345 968.061 985.868 1003.763

9.553 9.546 9.541 9.536 9.533

5600 5700 5800 5900 6000

93.525 93.706 93.883 94.057 94.228

412.014 413.671 415.302 416.909 418.491

333.845 335.231 336.597 337.945 339.274

437.748 447.110 456.489 465.886 475.301

12.189 7.303 2.393 2.537 7.488

1021.748 1039.819 1057.976 1076.217 1094.543

9.530 9.529 9.528 9.528 9.529

597

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Ethene (C2H4), ideal gas, molecular weight  28.05376 Enthalpy reference temperature  Tr  298.15 K [G°  H°(Tr)]/T (J/K/mol)

C°p (J/K/mol)

S° (J/K/mol)

0 100 200 250

0. 33.270 35.359 38.645

0. 180.542 203.955 212.172

Infinite 252.466 222.975 220.011

298.15

42.886

219.330

300 350 400 450 500

43.063 48.013 53.048 57.907 62.477

219.596 226.602 233.343 239.874 246.215

600 700 800 900 1000

70.663 77.714 83.840 89.200 93.899

1100 1200 1300 1400 1500

T (K)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

10.518 7.192 3.804 1.960

60.986 58.194 55.542 54.002

60.986 60.476 63.749 65.976

Infinite 31.589 16.649 13.785

219.330

0.

52.467

68.421

11.987

219.331 219.873 221.138 222.858 224.879

0.079 2.355 4.882 7.657 10.668

52.408 50.844 49.354 47.951 46.641

68.521 71.330 74.360 77.571 80.933

11.930 10.645 9.710 9.004 8.455

258.348 269.783 280.570 290.761 300.408

229.456 234.408 239.511 244.644 249.742

17.335 24.763 32.847 41.505 50.665

44.294 42.300 40.637 39.277 38.183

88.017 95.467 103.180 111.082 119.122

7.663 7.124 6.737 6.447 6.222

98.018 101.626 104.784 107.550 109.974

309.555 318.242 326.504 334.372 341.877

254.768 259.698 264.522 269.233 273.827

60.266 70.252 80.576 91.196 102.074

37.318 36.645 36.129 35.742 35.456

127.259 135.467 143.724 152.016 160.331

6.043 5.897 5.775 5.672 5.583

1600 1700 1800 1900 2000

112.103 113.976 115.628 117.089 118.386

349.044 355.898 362.460 368.752 374.791

278.306 282.670 286.922 291.064 295.101

113.181 124.486 135.968 147.606 159.381

35.249 35.104 35.005 34.938 34.894

168.663 177.007 185.357 193.712 202.070

5.506 5.439 5.379 5.326 5.278

2100 2200 2300 2400 2500

119.540 120.569 121.491 122.319 123.064

380.596 386.181 391.561 396.750 401.758

299.035 302.870 306.610 310.258 313.818

171.278 183.284 195.388 207.580 219.849

34.864 34.839 34.814 34.783 34.743

210.429 218.790 227.152 235.515 243.880

5.234 5.195 5.159 5.126 5.096

2600 2700 2800 2900 3000

123.738 124.347 124.901 125.404 125.864

406.596 411.280 415.812 420.204 424.463

317.294 320.689 324.006 327.248 330.418

232.190 244.595 257.058 269.573 282.137

34.688 34.616 34.524 34.409 34.269

252.246 260.615 268.987 277.363 285.743

5.068 5.042 5.018 4.996 4.975

3100 3200 3300 3400 3500

126.284 126.670 127.024 127.350 127.650

428.597 432.613 436.516 440.313 444.009

333.518 336.553 339.523 342.432 345.281

294.745 307.393 320.078 332.797 345.547

34.102 33.906 33.679 33.420 33.127

294.128 302.518 310.916 319.321 327.734

4.956 4.938 4.921 4.906 4.891

3600 3700 3800 3900 4000

127.928 128.186 128.424 128.646 128.852

447.609 451.118 454.539 457.878 461.138

348.074 350.812 353.497 356.130 358.715

358.326 371.132 383.962 396.816 409.691

32.800 32.436 32.035 31.596 31.118

336.156 344.588 353.030 361.482 369.947

4.877 4.865 4.853 4.842 4.831

598

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds

Ethene (C2H4), ideal gas, molecular weight  28.05376 Enthalpy reference temperature  Tr  298.15 K [G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

T (K)

C°p (J/K/mol)

S° (J/K/mol)

4100 4200 4300 4400 4500

129.045 129.224 129.392 129.549 129.696

464.322 467.434 470.476 473.453 476.366

361.252 363.743 366.190 368.594 370.957

422.586 435.500 448.430 461.378 474.340

30.600 30.041 29.441 28.799 28.116

378.424 388.915 395.418 403.937 412.470

4.821 4.812 4.803 4.795 4.788

4600 4700 4800 4900 5000

129.835 129.965 130.087 130.202 130.311

479.218 482.012 484.749 487.433 490.064

373.280 375.563 377.810 380.020 382.194

487.317 500.307 513.309 526.324 539.349

27.390 26.623 25.813 24.962 24.069

421.019 429.584 438.167 446.766 455.384

4.781 4.774 4.768 4.763 4.757

5100 5200 5300 5400 5500

130.413 130.510 130.602 130.689 130.771

492.646 495.179 497.666 500.108 502.507

384.335 386.442 388.517 390.561 392.575

552.386 565.432 578.488 591.552 604.625

23.136 22.162 21.149 20.097 19.008

464.019 472.673 481.346 490.040 498.751

4.753 4.748 4.744 4.740 4.737

5600 5700 5800 5900 6000

130.849 130.923 130.993 131.060 131.124

504.864 507.180 509.458 511.698 513.901

394.559 396.515 398.442 400.343 402.217

617.706 630.795 643.891 656.993 670.103

17.884 16.724 15.531 14.306 13.051

507.485 516.238 525.012 533.806 542.621

4.734 4.731 4.728 4.726 4.724

log Kf

TABLE A2 Thermochemical Data of Selected Chemical Compounds Hydrogen, monatomic (H), ideal gas, molecular weight  1.00794 Enthalpy reference temperature  Tr  298.15 K

[G°  H°(Tr)]/T H°  H°(Tr) (J/K/mol) (kJ/mol)

C°p (J/K/mol)

S° (J/K/mol)

0 100 200 250

0. 20.786 20.786 20.786

0. 92.009 106.417 111.055

Infinite 133.197 116.618 115.059

298.15

20.786

114.716

300 350 400 450 500

20.786 20.786 20.786 20.786 20.786

114.845 118.049 120.825 123.273 125.463

T (K)

Standard state pressure  p°  0.1 MPa ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

log Kf

6.197 4.119 2.040 1.001

216.035 216.614 217.346 217.687

216.035 212.450 208.004 205.629

Infinite 110.972 54.325 42.964

114.716

0.

217.999

203.278

35.613

114.717 114.970 115.532 116.259 117.072

0.038 1.078 2.117 3.156 4.196

218.011 218.326 218.637 218.946 219.254

203.186 200.690 198.150 195.570 192.957

35.378 29.951 25.876 22.701 20.158

599

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Hydrogen, monatomic (H), ideal gas, molecular weight  1.00794 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

[G°  H°(Tr)]/T H°  H°(Tr) (J/K/mol) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

T (K)

C°p (J/K/mol)

S° (J/K/mol)

600 700 800 900 1000

20.786 20.786 20.788 20.786 20.786

129.253 132.457 135.232 137.681 139.871

118.796 120.524 122.193 123.781 125.282

6.274 8.353 10.431 12.510 14.589

219.868 220.478 221.080 221.671 222.248

187.640 182.220 176.713 171.132 165.485

16.335 13.597 11.538 9.932 8.644

1100 1200 1300 1400 1500

20.786 20.786 20.786 20.786 20.786

141.852 143.660 145.324 146.865 148.299

126.700 128.039 129.305 130.505 131.644

16.667 18.746 20.824 22.903 24.982

222.807 223.346 223.865 224.361 224.836

159.782 154.028 148.230 142.394 136.522

7.587 6.705 5.956 5.313 4.754

1600 1700 1800 1900 2000

20.786 20.786 20.786 20.786 20.786

149.640 150.900 152.088 153.212 154.278

132.728 133.760 134.745 135.688 136.591

27.060 29.139 31.217 33.296 35.375

225.289 225.721 226.132 226.525 226.898

130.620 124.689 118.734 112.757 106.760

4.264 3.831 3.446 3.100 2.788

2100 2200 2300 2400 2500

20.786 20.786 20.786 20.786 20.786

155.293 156.260 157.184 158.068 158.917

137.458 138.291 139.092 139.864 140.610

37.453 39.532 41.610 43.689 45.768

227.254 227.593 227.916 228.224 228.518

100.744 94.712 88.664 82.603 76.530

2.506 2.249 2.014 1.798 1.599

2600 2700 2800 2900 3000

20.786 20.786 20.786 20.786 20.786

159.732 160.516 161.272 162.002 162.706

141.330 142.026 142.700 143.353 143.986

47.846 49.925 52.004 54.082 56.161

228.798 229.064 229.318 229.560 229.790

70.444 64.349 58.243 52.129 46.007

1.415 1.245 1.087 0.939 0.801

3100 3200 3300 3400 3500

20.786 20.786 20.786 20.786 20.786

163.388 164.048 164.688 165.308 165.911

144.601 145.199 145.780 146.345 146.895

58.239 60.318 62.397 64.475 66.554

230.008 230.216 230.413 230.599 230.776

39.877 33.741 27.598 21.449 15.295

0.672 0.551 0.437 0.330 0.228

3600 3700 3800 3900 4000

20.786 20.786 20.786 20.786 20.786

166.496 167.066 167.620 168.160 168.686

147.432 147.955 148.465 148.963 149.450

68.632 70.711 72.790 74.868 76.947

230.942 231.098 231.244 231.381 231.509

9.136 2.973 3.195 9.366 15.541

0.133 0.042 0.044 0.125 0.203

4100 4200 4300 4400 4500

20.786 20.786 20.786 20.786 20.786

169.200 169.700 170.190 170.667 171.135

149.925 150.390 150.845 151.290 151.726

79.025 81.104 83.183 85.261 87.340

231.627 231.736 231.836 231.927 232.009

21.718 27.899 34.082 40.267 46.454

0.277 0.347 0.414 0.478 0.539

4600 4700 4800 4900 5000

20.786 20.786 20.786 20.786 20.786

171.591 172.038 172.476 172.905 173.325

152.153 152.571 152.981 153.383 153.778

89.418 91.497 93.576 95.654 97.733

232.082 232.147 232.204 232.253 232.294

52.643 58.834 65.025 71.218 77.412

0.598 0.654 0.708 0.759 0.809

log Kf

600

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Hydrogen, monatomic (H), ideal gas, molecular weight  1.00794 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

[G°  H°(Tr)]/T H°  H°(Tr) (J/K/mol) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

T (K)

C°p (J/K/mol)

S° (J/K/mol)

5100 5200 5300 5400 5500

20.786 20.786 20.786 20.786 20.786

173.736 174.140 174.536 174.924 175.306

154.165 154.546 154.919 155.286 155.646

99.811 101.890 103.969 106.047 108.126

232.327 232.353 232.373 232.386 232.392

83.606 89.801 95.997 102.192 108.389

0.856 0.902 0.946 0.989 1.029

5600 5700 5800 5900 6000

20.786 20.786 20.786 20.786 20.786

175.680 176.048 176.410 176.765 177.114

156.001 156.349 156.692 157.029 157.361

110.204 112.283 114.362 116.440 118.519

232.393 232.389 232.379 232.365 232.348

114.584 120.780 126.976 133.172 139.368

1.069 1.107 1.144 1.179 1.213

log Kf

TABLE A2 Thermochemical Data of Selected Chemical Compounds Hydroxyl (OH), ideal gas, molecular weight  17.0074 Enthalpy reference temperature  Tr  298.15 K

[G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (kJ/mol) (kJ/mol)

C°p (J/K/mol)

S° (J/K/mol)

0 100 200 250

0. 32.627 30.777 30.283

0. 149.590 171.592 178.402

Infinite 210.980 186.471 184.204

298.15

29.986

183.708

300 350 400 450 500

29.977 29.780 29.650 29.567 29.521

183.894 188.499 192.466 195.954 199.066

600 700 800 900 1000

29.527 29.663 29.917 30.264 30.676

1100 1200 1300 1400 1500

31.124 31.586 32.046 32.492 32.917

T (K)

Standard state pressure  p°  0.1 MPa ΔfG° (kJ/mol)

log Kf

9.172 6.139 2.976 1.450

38.390 38.471 38.832 38.930

38.390 37.214 35.803 35.033

Infinite 19.438 9.351 7.320

183.708

0.

38.987

34.277

6.005

183.709 184.073 184.880 185.921 187.082

0.055 1.549 3.035 4.515 5.992

38.988 39.019 39.029 39.020 38.995

34.248 33.455 32.660 31.864 31.070

5.963 4.993 4.265 3.699 3.246

204.447 209.007 212.983 216.526 219.736

189.542 192.005 194.384 196.651 198.801

8.943 11.902 14.880 17.888 20.935

38.902 38.764 38.598 38.416 38.230

29.493 27.935 26.399 24.884 23.391

2.568 2.085 1.724 1.444 1.222

222.680 225.408 227.955 230.346 232.602

200.840 202.775 204.615 206.368 208.043

24.024 27.160 30.342 33.569 36.839

38.046 37.867 37.697 37.535 37.381

21.916 20.458 19.014 17.583 16.163

1.041 0.891 0.764 0.656 0.563

601

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Hydroxyl (OH), ideal gas, molecular weight  17.0074 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

[G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

T (K)

C°p (J/K/mol)

S° (J/K/mol)

1600 1700 1800 1900 2000

33.319 33.694 34.044 34.369 34.670

234.740 236.771 238.707 240.557 242.327

209.645 211.182 212.657 214.078 215.446

40.151 43.502 46.889 50.310 53.762

37.234 37.093 36.955 36.819 36.685

14.753 13.352 11.960 10.575 9.197

0.482 0.410 0.347 0.291 0.240

2100 2200 2300 2400 2500

34.950 35.209 35.449 35.673 35.881

244.026 245.658 247.228 248.741 250.202

216.767 218.043 219.278 220.474 221.635

57.243 60.752 64.285 67.841 71.419

36.551 36.416 36.278 36.137 35.992

7.826 6.462 5.103 3.750 2.404

0.195 0.153 0.116 0.082 0.050

2600 2700 2800 2900 3000

36.075 36.256 36.426 36.586 36.736

251.613 252.978 254.300 255.581 256.824

222.761 223.855 224.918 225.954 226.962

75.017 78.633 82.267 85.918 89.584

35.843 35.689 35.530 35.365 35.194

1.063 0.271 1.600 2.924 4.241

0.021 0.005 0.030 0.053 0.074

3100 3200 3300 3400 3500

36.878 37.013 37.140 37.261 37.376

258.031 259.203 260.344 261.455 262.537

227.945 228.904 229.839 230.753 231.645

93.265 96.960 100.667 104.387 108.119

35.017 34.834 34.644 34.448 34.246

5.552 6.858 8.158 9.452 10.741

0.094 0.112 0.129 0.145 0.160

3600 3700 3800 3900 4000

37.486 37.592 37.693 37.791 37.885

263.591 264.620 265.624 266.604 267.562

232.518 233.372 234.208 235.026 235.827

111.863 115.617 119.381 123.155 126.939

34.037 33.821 33.599 33.371 33.136

12.023 13.300 14.570 15.834 17.093

0.174 0.188 0.200 0.212 0.223

4100 4200 4300 4400 4500

37.976 38.064 38.150 38.233 38.315

268.499 269.415 270.311 271.189 272.050

236.613 237.383 238.138 238.879 239.607

130.732 134.534 138.345 142.164 145.991

32.894 32.646 32.391 32.130 31.862

18.346 19.593 20.833 22.068 23.297

0.234 0.244 0.253 0.262 0.270

4600 4700 4800 4900 5000

38.394 38.472 38.549 38.625 38.699

272.893 273.719 274.530 275.326 276.107

240.322 241.023 241.713 242.391 243.057

149.827 153.670 157.521 161.380 165.246

31.587 31.305 31.017 30.722 30.420

24.520 25.737 26.947 28.152 29.350

0.278 0.286 0.293 0.300 0.307

5100 5200 5300 5400 5500

38.773 38.846 38.919 38.991 39.062

276.874 277.627 278.368 279.096 279.812

243.713 244.358 244.993 245.617 246.233

169.120 173.001 176.889 180.784 184.687

30.111 29.796 29.473 29.144 28.807

30.542 31.729 32.909 34.083 35.251

0.313 0.319 0.324 0.330 0.335

5600 5700 5800 5900 6000

39.134 39.206 39.278 39.350 39.423

280.517 281.210 281.892 282.564 283.226

246.839 247.436 248.024 248.604 249.175

188.597 192.514 196.438 200.369 204.308

28.464 28.113 27.756 27.391 27.019

36.412 37.568 38.716 39.860 40.997

0.340 0.344 0.349 0.353 0.357

log Kf

602

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Hydroperoxyl (HO2), ideal gas, molecular weight  33.00674 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

10.003 6.677 3.346 1.659

5.006 3.928 3.001 2.532

ΔfG° (kJ/mol) 5.006 7.052 10.536 12.475

log Kf

0 100 200 250

0. 33.258 33.491 34.044

0. 192.430 215.515 223.040

Infinite 259.201 232.243 229.676

Infinite 3.683 2.752 2.606

298.15

34.905

229.106

229.106

0.

2.092

14.430

2.528

300 350 400 450 500

34.943 36.072 37.296 38.519 39.687

229.322 234.791 239.688 244.151 248.271

229.107 229.536 230.504 231.776 233.222

0.065 1.839 3.673 5.569 7.524

2.076 1.649 1.260 0.908 0.591

14.506 16.612 18.777 20.988 23.236

2.526 2.479 2.452 2.436 2.427

600 700 800 900 1000

41.781 43.558 45.084 46.418 47.604

255.697 262.275 268.193 273.582 278.535

236.363 239.603 242.813 245.937 248.952

11.601 15.870 20.304 24.880 29.583

0.043 0.411 0.790 1.107 1.368

27.819 32.485 37.211 41.981 46.783

2.422 2.424 2.430 2.436 2.444

1100 1200 1300 1400 1500

48.672 49.643 50.535 51.360 52.128

283.123 287.400 291.410 295.185 298.755

251.853 254.639 257.315 259.886 262.360

34.397 39.314 44.323 49.419 54.593

1.582 1.754 1.887 1.988 2.058

51.609 56.452 61.308 66.173 71.045

2.451 2.457 2.463 2.469 2.474

1600 1700 1800 1900 2000

52.845 53.518 54.149 54.742 55.299

302.143 305.367 308.444 311.388 314.210

264.741 267.037 269.252 271.393 273.464

59.842 65.161 70.545 75.989 81.492

2.102 2.122 2.121 2.102 2.067

75.920 80.797 85.674 90.551 95.427

2.479 2.483 2.486 2.489 2.492

2100 2200 2300 2400 2500

55.820 56.308 56.763 57.186 57.578

316.921 319.529 322.042 324.467 326.809

275.469 277.413 279.299 281.131 282.911

87.048 92.655 98.309 104.006 109.745

2.019 1.960 1.893 1.818 1.740

100.301 105.172 110.040 114.905 119.767

2.495 2.497 2.499 2.501 2.502

2600 2700 2800 2900 3000

57.940 58.274 58.580 58.860 59.115

329.075 331.268 333.393 335.453 337.453

284.644 286.330 287.973 289.575 291.138

115.521 121.332 127.175 133.047 138.946

1.659 1.577 1.497 1.419 1.346

124.625 129.481 134.334 139.183 144.030

2.504 2.505 2.506 2.507 2.508

3100 3200 3300 3400 3500

59.347 59.557 59.745 59.915 60.066

339.395 341.283 343.118 344.904 346.643

292.663 294.153 295.609 297.033 298.426

144.869 150.814 156.780 162.763 168.762

1.278 1.218 1.165 1.122 1.089

148.876 153.718 158.558 163.398 168.236

2.509 2.509 2.510 2.510 2.511

3600 3700 3800

60.200 60.318 60.423

348.337 349.989 351.599

299.789 301.123 302.431

174.775 180.801 186.839

1.067 1.058 1.061

173.074 177.910 182.748

2.511 2.512 2.512

603

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Hydroperoxyl (HO2), ideal gas, molecular weight  33.00674 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

log Kf

3900 4000

60.513 60.592

353.169 354.702

303.711 304.987

192.886 198.941

1.078 1.109

187.586 192.423

2.512 2.513

4100 4200 4300 4400 4500

60.659 60.716 60.764 60.803 60.834

356.199 357.662 359.091 360.488 361.855

306.199 307.406 308.592 309.756 310.898

205.004 211.072 217.146 223.225 229.307

1.156 1.217 1.296 1.391 1.504

197.262 202.102 206.945 211.788 216.634

2.513 2.514 2.514 2.514 2.515

4600 4700 4800 4900 5000

60.858 60.876 60.888 60.895 60.897

363.193 364.502 365.783 367.039 368.269

312.021 313.123 314.207 315.272 316.320

235.391 241.478 247.566 253.656 259.745

1.634 1.784 1.952 2.141 2.350

221.482 226.335 231.190 236.050 240.913

2.515 2.515 2.516 2.516 2.517

5100 5200 5300 5400 5500

60.894 60.888 60.879 60.866 60.851

369.475 370.658 371.817 372.955 374.072

317.351 318.364 319.362 320.344 321.311

265.835 271.924 278.012 284.100 290.185

2.580 2.831 3.105 3.402 3.722

245.780 250.652 255.530 260.412 265.300

2.517 2.518 2.518 2.519 2.520

5600 5700 5800 5900 6000

60.834 60.814 60.792 60.769 60.745

375.168 376.245 377.302 378.341 379.362

322.263 323.200 324.124 325.034 325.931

296.270 302.352 308.432 314.511 320.586

4.067 4.437 4.832 5.254 5.702

270.195 275.096 280.004 284.917 289.839

2.520 2.521 2.522 2.522 2.523

TABLE A2 Thermochemical Data of Selected Chemical Compounds Hydrogen (H2), ideal gas-reference state, molecular weight  2.01588 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

log Kf

8.467 5.468 2.774 1.378

0. 0. 0. 0.

0. 0. 0. 0.

0. 0. 0. 0.

130.680

0.

0.

0.

0.

130.680 131.032

0.053 1.502

0. 0.

0. 0.

0. 0.

0 100 200 250

0. 28.154 27.447 28.344

0. 100.727 119.412 125.640

Infinite 155.408 133.284 131.152

298.15

28.836

130.680

300 350

28.849 29.081

130.858 135.325

604

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Hydrogen (H2), ideal gas-reference state, molecular weight  2.01588 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

log Kf

400 450 500

29.181 29.229 29.260

139.216 142.656 145.737

131.817 132.834 133.973

2.959 4.420 5.882

0. 0. 0.

0. 0. 0.

0. 0. 0.

600 700 800 900 1000

29.327 29.441 29.624 29.881 30.205

151.077 155.606 159.548 163.051 166.216

136.392 138.822 141.171 143.411 145.536

8.811 11.749 14.702 17.676 20.680

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

1100 1200 1300 1400 1500

30.581 30.992 31.423 31.861 32.298

169.112 171.790 174.288 176.633 178.846

147.549 149.459 151.274 153.003 154.652

23.719 26.797 29.918 33.082 36.290

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

1600 1700 1800 1900 2000

32.725 33.139 33.537 33.917 34.280

180.944 182.940 184.846 186.669 188.418

156.231 157.743 159.197 160.595 161.943

39.541 42.835 46.169 49.541 52.951

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

2100 2200 2300 2400 2500

34.624 34.952 35.263 35.559 35.842

190.099 191.718 193.278 194.785 196.243

163.244 164.501 165.719 166.899 168.044

56.397 59.876 63.387 66.928 70.498

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

2600 2700 2800 2900 3000

36.111 36.370 36.618 36.856 37.087

197.654 199.021 200.349 201.638 202.891

169.155 170.236 171.288 172.313 173.311

74.096 77.720 81.369 85.043 88.740

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

3100 3200 3300 3400 3500

37.311 37.528 37.740 37.946 38.149

204.111 205.299 206.457 207.587 208.690

174.285 175.236 176.164 177.072 177.960

92.460 96.202 99.966 103.750 107.555

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

3600 3700 3800 3900 4000

38.348 38.544 38.738 38.928 39.116

209.767 210.821 211.851 212.860 213.848

178.828 179.679 180.512 181.328 182.129

111.380 115.224 119.089 122.972 126.874

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

4100 4200 4300 4400 4500

39.301 39.484 39.665 39.842 40.017

214.816 215.765 216.696 217.610 218.508

182.915 183.686 184.442 185.186 185.916

130.795 134.734 138.692 142.667 146.660

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

605

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Hydrogen (H2), ideal gas-reference state, molecular weight  2.01588 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

log Kf

4600 4700 4800 4900 5000

40.188 40.355 40.518 40.676 40.829

219.389 220.255 221.106 221.943 222.767

186.635 187.341 188.035 188.719 189.392

150.670 154.698 158.741 162.801 166.876

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

5100 5200 5300 5400 5500

40.976 41.117 41.252 41.379 41.498

223.577 224.374 225.158 225.931 226.691

190.054 190.706 191.349 191.982 192.606

170.967 175.071 179.190 183.322 187.465

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

5600 5700 5800 5900 6000

41.609 41.712 41.806 41.890 41.965

227.440 228.177 228.903 229.619 230.323

193.222 193.829 194.427 195.017 195.600

191.621 195.787 199.963 204.148 208.341

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

TABLE A2 Thermochemical Data of Selected Chemical Compounds Water (H2O), ideal gas, molecular weight  18.01528 Enthalpy reference temperature  Tr  298.15 K

T (K) 0 100 200

C°p (J/K/mol) 0. 33.299 33.349

Standard state pressure  p°  0.1 MPa

S° [G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (J/K/mol) (kJ/mol) (kJ/mol) 0. 152.388 175.485

Infinite 218.534 191.896

9.904 6.615 3.282

238.921 240.083 240.900

ΔfG° (kJ/mol) 238.921 236.584 232.766

log Kf Infinite 123.579 60.792

298.15

33.590

188.834

188.834

0.

241.826

228.582

40.047

300 400 500

33.596 34.262 35.226

189.042 198.788 206.534

188.835 190.159 192.685

0.062 3.452 6.925

241.844 242.846 243.826

228.500 223.901 219.051

39.785 29.238 22.884

600 700 800 900 1000

36.325 37.495 38.721 39.987 41.268

213.052 218.739 223.825 228.459 232.738

195.550 198.465 201.322 204.084 206.738

10.501 14.192 18.002 21.938 26.000

244.758 245.632 246.443 247.185 247.857

214.007 208.612 203.496 198.083 192.590

18.631 15.582 13.287 11.496 10.060

1100 1200 1300

42.536 43.768 44.945

236.731 240.485 244.035

209.285 211.730 214.080

30.191 34.506 38.942

248.460 248.997 249.473

187.033 181.425 175.774

8.881 7.897 7.063

606

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Water (H2O), ideal gas, molecular weight  18.01528 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

S° [G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (J/K/mol) (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

T (K)

C°p (J/K/mol)

1400 1500

46.054 47.090

247.407 250.620

216.341 218.520

43.493 48.151

249.894 250.265

170.089 164.376

6.346 5.724

1600 1700 1800 1900 2000

48.050 48.935 49.749 50.496 51.180

253.690 256.630 259.451 262.161 264.769

220.623 222.655 224.621 226.526 228.374

52.908 57.758 62.693 67.706 72.790

250.592 250.881 251.138 251.368 251.575

158.639 152.883 147.111 141.325 135.528

5.179 4.698 4.269 3.885 3.540

2100 2200 2300 2400 2500

51.823 52.408 52.947 53.444 53.904

267.282 269.706 272.048 274.312 276.503

230.167 231.909 233.604 235.253 236.860

77.941 83.153 88.421 93.741 99.108

251.762 251.934 252.092 252.239 252.379

129.721 123.905 118.082 112.252 106.416

3.227 2.942 2.682 2.443 2.223

2600 2700 2800 2900 3000

54.329 54.723 55.089 55.430 55.748

278.625 280.683 282.680 284.619 286.504

238.425 239.952 241.443 242.899 244.321

104.520 109.973 115.464 120.990 126.549

252.513 252.643 252.771 252.897 253.024

100.575 94.729 88.878 83.023 77.163

2.021 1.833 1.658 1.495 1.344

3100 3200 3300 3400 3500

56.044 56.323 56.583 56.828 57.058

288.337 290.120 291.858 293.550 295.201

245.711 247.071 248.402 249.705 250.982

132.139 137.757 143.403 149.073 154.768

253.152 253.282 253.416 253.553 253.696

71.298 65.430 59.558 53.681 47.801

1.201 1.068 0.943 0.825 0.713

3600 3700 3800 3900 4000

57.276 57.480 57.675 57.859 58.033

296.812 298.384 299.919 301.420 302.887

252.233 253.459 254.661 255.841 256.999

160.485 166.222 171.980 177.757 183.552

253.844 253.997 254.158 254.326 254.501

41.916 36.027 30.133 24.236 18.334

0.608 0.509 0.414 0.325 0.239

4100 4200 4300 4400 4500

58.199 58.357 58.507 58.650 58.787

304.322 305.726 307.101 308.448 309.767

258.136 259.252 260.349 261.427 262.486

189.363 195.191 201.034 206.892 212.764

254.684 254.876 255.078 255.288 255.508

12.427 6.516 0.600 5.320 11.245

0.158 0.081 0.007 0.063 0.131

4600 4700 4800 4900 5000

58.918 59.044 59.164 59.275 59.390

311.061 312.329 313.574 314.795 315.993

263.528 264.553 265.562 266.554 267.531

218.650 224.548 230.458 236.380 242.313

255.738 255.978 256.229 256.491 256.763

17.175 23.111 29.052 34.998 40.949

0.195 0.257 0.316 0.373 0.428

5100 5200 5300 5400 5500

59.509 59.628 59.746 59.864 59.982

317.171 318.327 319.464 320.582 321.682

268.493 269.440 270.373 271.293 272.199

248.256 254.215 260.184 266.164 272.157

257.046 257.338 257.639 257.950 258.268

46.906 52.869 58.838 64.811 70.791

0.480 0.531 0.580 0.627 0.672

log Kf

607

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Water (H2O), ideal gas, molecular weight  18.01528 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p (J/K/mol)

5600 5700 5800 5900 6000

60.100 60.218 60.335 60.453 60.571

Standard state pressure  p°  0.1 MPa

S° [G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (J/K/mol) (kJ/mol) (kJ/mol) 322.764 323.828 324.877 325.909 326.926

273.092 273.973 274.841 275.698 276.544

278.161 284.177 290.204 296.244 302.295

258.595 258.930 259.272 259.621 259.977

ΔfG° (kJ/mol) 76.777 82.769 88.787 94.770 100.780

log Kf 0.716 0.758 0.799 0.839 0.877

TABLE A2 Thermochemical Data of Selected Chemical Compounds Ammonia (NH3), ideal gas, molecular weight  17.0352 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

log Kf

0 100 200

0. 33.284 33.757

0. 155.840 178.990

Infinite 223.211 195.962

10.045 6.737 3.394

38.907 41.550 43.703

38.907 34.034 25.679

Infinite 17.777 6.707

298.15 300 400 500

35.652 35.701 38.716 42.048

192.774 192.995 203.663 212.659

192.774 192.775 194.209 197.021

0. 0.066 3.781 7.819

45.898 45.939 48.041 49.857

16.367 16.183 5.941 4.800

2.867 2.818 0.776 0.501

600 700 800 900 1000

45.293 48.354 51.235 53.948 56.491

220.615 227.829 234.476 240.669 246.486

200.302 203.727 207.160 210.543 213.849

12.188 16.872 21.853 27.113 32.637

51.374 52.618 53.621 54.411 55.013

15.879 27.190 38.662 50.247 61.910

1.382 2.029 2.524 2.916 3.234

1100 1200 1300 1400 1500

58.859 61.048 63.057 64.893 66.564

251.983 257.199 262.166 266.907 271.442

217.069 220.197 223.236 226.187 229.054

38.406 44.402 50.609 57.008 63.582

55.451 55.746 55.917 55.982 55.954

73.625 85.373 97.141 108.918 120.696

3.496 3.716 3.903 4.064 4.203

1600 1700 1800 1900 2000

68.079 69.452 70.695 71.818 72.833

275.788 279.957 283.962 287.815 291.525

231.840 234.549 237.184 239.748 242.244

70.315 77.193 84.201 91.328 98.561

55.847 55.672 55.439 55.157 54.833

132.469 144.234 155.986 167.725 179.447

4.325 4.432 4.527 4.611 4.687

2100 2200 2300

73.751 74.581 75.330

295.101 298.552 301.884

244.677 247.048 249.360

105.891 113.309 120.805

54.473 54.084 53.671

191.152 202.840 214.509

4.755 4.816 4.872

608

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Ammonia (NH3), ideal gas, molecular weight  17.0352 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

log Kf

2400 2500

76.009 76.626

305.104 308.220

251.616 253.818

128.372 136.005

53.238 52.789

228.160 237.792

4.922 4.968

2600 2700 2800 2900 3000

77.174 77.672 78.132 78.529 78.902

311.236 314.158 316.991 319.740 322.409

255.969 258.070 260.124 262.132 264.097

143.695 151.438 159.228 167.062 174.933

52.329 51.860 51.386 50.909 50.433

249.406 261.003 272.581 284.143 295.689

5.011 5.049 5.085 5.118 5.148

3100 3200 3300 3400 3500

79.228 79.521 79.785 80.011 80.216

325.001 327.521 329.972 332.358 334.680

266.020 267.903 269.747 271.554 273.324

182.840 190.778 198.744 206.734 214.745

49.959 49.491 49.030 48.578 48.139

307.218 318.733 330.233 341.719 353.191

5.177 5.203 5.227 5.250 5.271

3600 3700 3800 3900 4000

80.400 80.550 80.684 80.793 80.881

336.942 339.147 341.297 343.395 345.441

275.060 276.763 278.433 280.072 281.680

222.776 230.824 238.886 246.960 255.043

47.713 47.302 46.908 46.534 46.180

364.652 376.101 387.539 398.967 410.385

5.291 5.310 5.327 5.344 5.359

4100 4200 4300 4400 4500

80.956 81.006 81.048 81.065 81.073

347.439 349.391 351.297 353.161 354.983

283.260 284.811 286.335 287.833 289.305

263.136 271.234 279.337 287.442 295.550

45.847 45.539 45.254 44.996 44.764

421.795 433.198 444.593 455.981 467.364

5.374 5.388 5.401 5.413 5.425

4600 4700 4800 4900 5000

81.057 81.032 80.990 80.931 80.856

356.765 358.508 360.213 361.882 363.517

290.752 292.175 293.575 294.952 296.307

303.656 311.761 319.862 327.958 336.048

44.561 44.387 44.242 44.129 44.047

478.743 490.117 501.488 512.856 524.223

5.436 5.447 5.457 5.467 5.477

5100 5200 5300 5400 5500

80.751 80.751 80.751 80.751 80.751

365.117 366.685 368.223 369.732 371.214

297.641 298.954 300.246 301.519 302.773

344.127 352.202 360.277 368.352 376.428

43.999 43.979 43.982 44.006 44.049

535.587 546.951 558.315 569.680 581.044

5.486 5.494 5.503 5.511 5.518

5600 5700 5800 5900 6000

80.751 80.751 80.751 80.751 80.751

372.669 374.098 375.503 376.883 378.240

304.008 305.225 306.425 307.607 308.773

384.503 392.578 400.653 408.728 416.803

44.112 44.193 44.291 44.404 44.531

592.410 603.778 615.147 626.516 637.889

5.526 5.533 5.540 5.547 5.553

609

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Magnesium (Mg), crystal–liquid, molecular weight  24.305 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

[G°  H°(Tr)]/T H°  H°(Tr) (J/K/mol) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

C°p (J/K/mol)

S° (J/K/mol)

0 100 200 250

0. 15.762 22.724 24.018

0. 9.505 23.143 28.364

Infinite 53.066 34.888 33.076

4.998 4.356 2.349 1.178

0. 0. 0. 0.

0. 0. 0. 0.

0. 0. 0. 0.

298.15

24.869

32.671

32.671

0.

0.

0.

0.

300 350 400 450 500

24.897 25.568 26.144 26.668 27.171

32.825 36.715 40.167 43.277 46.113

32.671 32.977 33.664 34.562 35.578

0.046 1.308 2.601 3.922 5.268

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

600 700 800 900

28.184 29.279 30.507 31.895

51.156 55.581 59.569 63.241

37.764 39.999 42.200 44.336

8.035 10.907 13.895 17.014

0. 0. 0. 0.

0. 0. 0. 0.

0. 0. 0. 0.

923 923

32.238 34.309

64.050 73.234

44.818 44.818

17.751 26.228

1000

34.309

75.983

47.113

28.870

0.

0.

0.

1100 1200 1300

34.309 34.309 34.309

79.253 82.238 84.984

49.888 52.462 54.859

32.301 35.732 39.163

0. 0. 0.

0. 0. 0.

0. 0. 0.

1366.104

34.309

86.686

56.358

41.431

1400 1500

34.309 34.309

87.527 89.894

57.103 59.211

42.594 46.024

127.409 126.057

3.167 12.447

0.118 0.433

1600 1700 1800 1900 2000

34.309 34.309 34.309 34.309 34.309

92.108 94.188 96.149 98.004 99.764

61.199 63.079 64.862 66.558 68.175

49.455 52.886 56.317 59.748 63.179

124.705 123.352 122.000 120.648 119.296

21.636 30.741 39.766 48.717 57.596

0.706 0.945 1.154 1.339 1.504

T (K)

log Kf

Crystal–liquid transition

Fugacity  1 bar

610

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds

Magnesium (Mg), ideal gas, molecular weight  44.0098 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa

[G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

6.197 4.119 2.040 1.001

145.901 147.337 147.409 147.277

145.901 135.694 123.968 118.122

Infinite 70.879 32.377 24.680

148.648

0.

147.100

112.522

19.713

148.648 148.901 149.463 150.190 151.003

0.038 1.078 2.117 3.156 4.196

147.092 146.870 146.616 146.335 146.028

112.307 106.527 100.780 95.067 89.387

19.554 15.898 13.161 11.035 9.338

163.184 166.388 169.164 171.612 173.802

152.727 154.455 156.124 157.712 159.213

6.274 8.353 10.431 12.510 14.589

145.339 144.546 143.636 142.596 132.819

78.122 66.981 55.961 45.062 35.000

6.801 4.998 3.654 2.615 1.828

175.783 177.592 179.255

160.631 161.970 163.237

16.667 18.746 20.824

131.466 130.114 128.762

25.283 15.690 6.209

1.201 0.683 0.249

0 100 200 250

0. 20.786 20.786 20.786

0. 125.940 140.348 140.986

Infinite 167.128 150.549 148.990

298.15

20.786

148.648

300 350 400 450 500

20.786 20.786 20.786 20.786 20.786

148.776 151.980 154.756 157.204 159.394

600 700 800 900 1000

20.786 20.786 20.786 20.786 20.786

1100 1200 1300

20.786 20.786 20.786

Fugacity  1 bar

1366.104

20.786

180.286

164.037

22.199

1400 1500

20.786 20.786

180.796 182.230

164.437 165.575

22.903 24.982

0. 0.

0. 0.

0. 0.

1600 1700 1800 1900 2000

20.786 20.786 20.787 20.787 20.789

183.571 184.832 186.020 187.144 188.210

166.659 167.691 168.677 169.619 170.522

27.060 29.139 31.218 33.296 35.375

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

2100 2200 2300 2400 2500

20.791 20.795 20.802 20.812 20.826

189.224 190.192 191.116 192.002 192.851

171.389 172.222 173.023 173.796 174.541

37.454 39.533 41.613 43.694 45.776

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

2600 2700 2800 2900 3000

20.846 20.874 20.909 20.956 21.014

193.669 194.456 195.216 195.950 196.661

175.261 175.958 176.632 177.285 177.920

47.859 49.945 52.034 54.127 56.226

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

3100 3200 3300 3400 3500

21.085 21.172 21.275 21.396 21.537

197.352 198.022 198.675 199.312 199.934

178.535 179.134 179.716 180.283 180.836

58.331 60.443 62.566 64.699 66.845

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

3600 3700

21.697 21.879

200.543 201.140

181.375 181.901

69.007 71.186

0. 0.

0. 0.

0. 0.

611

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds

Magnesium (Mg), ideal gas, molecular weight  44.0098 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

Standard state pressure  p°  0.1 MPa

[G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

3800 3900 4000

22.083 22.310 22.559

201.726 202.303 202.871

182.415 182.918 183.409

73.384 75.603 77.846

0. 0. 0.

0. 0. 0.

0. 0. 0.

4100 4200 4300 4400 4500

22.832 23.128 23.447 23.789 24.152

203.431 203.985 204.533 205.076 205.614

183.891 184.363 184.825 185.279 185.725

80.116 82.413 84.742 87.103 89.500

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

4600 4700 4800 4900 5000

24.537 24.944 25.372 25.820 26.287

206.149 206.681 207.211 207.739 208.265

186.164 186.594 187.018 187.436 187.847

91.934 94.408 96.923 99.483 102.088

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

5100 5200 5300 5400 5500

26.773 27.276 27.794 28.329 28.878

208.790 209.315 209.839 210.364 210.888

188.253 188.653 189.048 189.437 189.823

104.741 107.443 110.196 113.002 115.862

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

5600 5700 5800 5900 6000

29.442 30.020 30.610 31.213 31.827

211.414 211.940 212.467 212.996 213.525

190.204 190.580 190.953 191.322 191.688

118.778 121.751 124.782 127.873 131.025

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

TABLE A2 Thermochemical Data of Selected Chemical Compounds Magnesium oxide (MgO), crystal–liquid, molecular weight  40.3044 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa

T (K)

C°p (J/K/mol)

S° [G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (J/K/mol) (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

0 100 200

0. 7.802 26.681

0. 2.548 14.096

Infinite 52.212 30.037

5.159 4.966 3.188

log Kf

597.060 598.962 600.646

597.060 589.601 579.488

Infinite 307.976 151.347

298.15

37.106

26.924

26.924

0.

601.241

568.945

99.677

300 400 500

37.244 42.561 45.544

27.154 38.678 48.523

26.925 28.460 31.513

0.069 4.087 8.505

601.245 601.268 601.046

568.745 557.898 547.078

99.027 72.854 57.153

612

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Magnesium oxide (MgO), crystal–liquid, molecular weight  40.3044 Enthalpy reference temperature  Tr  298.15 K Standard state pressure  p°  0.1 MPa S° [G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (J/K/mol) (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

T (K)

C°p (J/K/mol)

600 700 800 900 1000

47.430 48.748 49.740 50.539 51.208

57.006 64.420 70.996 76.902 82.262

35.072 38.746 42.374 45.888 49.262

13.160 17.972 22.898 27.913 33.001

600.738 600.426 600.156 599.962 608.462

536.312 525.600 514.930 504.289 492.952

46.690 39.221 33.621 29.268 25.749

1100 1200 1300 1400 1500

51.794 52.325 52.810 53.262 53.693

87.171 91.701 95.908 99.839 103.528

52.488 55.569 58.512 61.325 64.017

38.151 43.358 48.615 53.918 59.266

608.496 608.495 608.461 735.804 734.355

481.399 469.844 458.291 443.575 422.752

22.880 20.452 18.414 16.550 14.722

1600 1700 1800 1900 2000

54.107 54.509 54.898 55.278 55.651

107.007 110.299 113.426 116.404 119.249

66.596 69.071 71.449 73.737 75.942

64.656 70.087 75.558 81.067 86.613

732.877 731.371 729.837 728.277 726.690

402.026 381.394 360.851 340.395 320.021

13.125 11.719 10.472 9.358 8.358

2100 2200 2300 2400 2500

56.019 56.379 56.738 57.094 57.445

121.973 124.587 127.101 129.524 131.862

78.070 80.125 82.114 84.039 85.905

92.197 97.816 103.472 109.164 114.891

725.079 723.442 721.781 720.097 718.389

299.727 279.510 259.368 239.300 219.301

7.455 6.636 5.890 5.208 4.582

2600 2700 2800 2900 3000

57.797 58.145 58.491 58.836 59.177

134.121 136.309 138.430 140.489 142.489

87.716 89.476 91.187 92.851 94.473

120.653 126.450 132.282 138.148 144.049

716.659 714.906 713.132 711.338 709.524

199.372 179.509 159.712 139.979 120.308

4.005 3.473 2.979 2.521 2.095

3100

59.516

144.436

96.053

149.987

707.689

100.697

1.697

3105 3105

59.523 66.944

144.532 169.595

96.131 96.131

150.284 228.107

3200 3300 3400 3500

66.944 66.944 66.944 66.944

171.613 173.673 175.671 177.612

98.342 100.594 102.773 104.884

234.466 241.161 247.855 254.550

627.329 624.772 622.235 619.719

83.538 66.584 49.708 32.905

1.364 1.054 0.764 0.491

3600 3700 3800 3900 4000

66.944 66.944 66.944 66.944 66.944

179.498 181.332 183.117 184.856 186.551

106.930 108.916 110.846 112.721 114.546

261.244 267.938 274.633 281.327 288.022

617.226 614.757 612.315 609.901 607.518

16.174 0.487 17.083 33.615 50.084

0.235 0.007 0.235 0.450 0.654

4100 4200 4300 4400 4500

66.944 66.944 66.944 66.944 66.944

188.204 189.817 191.393 192.932 194.436

116.322 118.053 119.740 121.386 122.993

294.716 301.410 308.105 314.799 321.494

605.167 602.851 600.572 598.332 596.133

66.495 82.849 99.149 115.395 131.591

0.847 1.030 1.204 1.370 1.527

4600 4700 4800 4900 5000

66.944 66.944 66.944 66.944 66.944

195.907 197.347 198.756 200.137 201.489

124.562 126.095 127.595 129.061 130.496

328.188 334.882 341.577 348.271 354.966

593.978 591.869 589.807 587.796 585.838

147.739 163.840 179.897 195.912 211.886

1.678 1.821 1.958 2.088 2.214

log Kf

Crystal–liquid transition

613

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Magnesium oxide (MgO), crystal–liquid, molecular weight  40.3044 Enthalpy reference temperature  Tr  298.15 K

[G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (kJ/mol) (kJ/mol)

C°p (J/K/mol)

S° (J/K/mol)

0 100 200 250

0. 29.124 30.117 31.121

0. 180.485 200.873 207.699

Infinite 240.493 216.143 213.793

298.15

32.173

213.269

300 350 400 450 500

32.215 33.401 34.783 36.452 38.434

213.468 218.522 223.069 227.259 231.200

600 700 800 900 1000

43.078 47.785 51.560 53.890 54.795

1100 1200 1300 1400 1500

T (K)

Standard state pressure  p°  0.1 MPa ΔfG° (kJ/mol)

log Kf

8.909 6.001 3.054 1.524

58.588 59.402 58.886 58.517

58.588 50.970 42.689 38.682

Infinite 26.624 11.149 8.082

213.269

0.

58.158

34.895

6.113

213.270 213.666 214.562 215.743 217.093

0.060 1.699 3.403 5.182 7.053

58.144 57.783 57.446 57.147 56.901

34.750 30.880 27.060 23.280 19.531

6.051 4.609 3.534 2.702 2.040

238.608 245.608 252.250 258.472 264.208

220.068 223.221 226.439 229.657 232.829

11.124 15.671 20.649 25.934 31.379

56.624 56.672 56.994 57.457 49.315

54.608 53.729 52.496 51.139 49.802

269.429 274.147 278.400 282.241 285.723

235.923 238.915 241.792 244.546 247.177

36.857 42.278 47.591 52.773 57.820

49.607 49.823 49.914 77.551 76.404

23.779 30.461 37.156 40.685 38.093

1.129 1.326 1.493 1.518 1.327

1600 1700 1800 1900 2000

48.558 47.439 46.454 45.599 44.862

288.897 291.807 294.490 296.978 299.298

249.687 252.080 254.363 256.541 258.621

62.737 67.535 72.229 76.831 81.353

75.399 74.525 73.768 73.114 72.552

35.573 33.111 30.697 28.323 25.980

1.161 1.017 0.891 0.779 0.679

2100 2200 2300 2400 2500

44.230 43.691 43.231 42.841 42.509

301.471 303.516 305.447 307.279 309.021

260.611 262.515 264.340 266.091 267.774

85.806 90.202 94.547 98.850 103.117

72.070 71.658 71.308 71.012 70.764

23.664 21.368 19.091 16.827 14.574

0.589 0.507 0.434 0.366 0.305

2600 2700 2800 2900 3000

42.229 41.993 41.795 41.629 41.493

310.682 312.272 313.795 315.259 316.668

269.392 270.951 272.454 273.905 275.307

107.354 111.565 115.754 119.925 124.081

70.559 70.393 70.262 70.163 70.094

12.331 10.095 7.864 5.637 3.414

0.248 0.195 0.147 0.102 0.059

3100 3200 3300 3400 3500

41.381 41.291 41.220 41.167 41.128

318.026 319.339 320.608 321.838 323.030

276.664 277.977 279.250 280.484 281.683

128.224 132.357 136.483 140.602 144.717

70.053 70.040 70.052 70.090 70.154

1.191 1.029 3.250 5.472 7.695

0.020 0.017 0.051 0.084 0.115

3600 3700 3800

41.103 41.091 41.090

324.189 325.315 326.411

282.848 283.980 285.082

148.828 152.938 157.047

70.243 70.359 70.503

9.921 12.149 14.381

0.144 0.172 0.198

12.088 1.052 4.666 0.348 2.783 0.182 10.282 0.597 17.121 0.894

614

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Magnesium oxide (MgO), crystal–liquid, molecular weight  40.3044 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

[G°  H°(Tr)]/T H°  H°(Tr) ΔfH° (J/K/mol) (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

T (K)

C°p (J/K/mol)

S° (J/K/mol)

3900 4000

41.099 41.117

327.478 328.519

286.156 287.202

161.156 165.267

70.674 70.874

16.617 18.858

0.223 0.246

4100 4200 4300 4400 4500

41.144 41.179 41.222 41.272 41.329

329.534 330.526 331.496 332.444 333.372

288.222 289.218 290.190 291.139 292.067

169.380 173.496 177.616 181.741 185.871

71.105 71.367 71.662 71.992 72.358

21.104 23.356 25.615 27.881 30.154

0.269 0.290 0.311 0.331 0.350

4600 4700 4800 4900 5000

41.393 41.463 41.539 41.621 41.709

334.281 335.172 336.046 336.903 337.745

292.975 293.864 294.733 295.585 296.420

190.007 194.149 198.300 202.458 206.624

72.761 73.203 73.686 74.212 74.781

32.437 34.729 37.030 39.342 41.665

0.368 0.386 0.403 0.419 0.435

5100 5200 5300 5400 5500

41.802 41.901 42.004 42.113 42.227

338.572 339.384 340.183 340.970 341.743

297.238 298.041 298.829 299.602 300.361

210.800 214.985 219.180 223.386 227.603

75.395 76.057 76.766 77.525 78.335

44.000 46.347 48.709 51.083 53.471

0.451 0.466 0.480 0.494 0.508

5600 5700 5800 5900 6000

42.345 42.467 42.594 42.725 42.860

342.505 343.256 343.995 344.725 345.444

301.107 301.840 302.560 303.269 303.966

231.831 236.072 240.325 244.591 248.870

79.198 80.115 81.087 82.116 83.202

55.876 58.296 60.733 63.186 65.658

0.521 0.534 0.547 0.559 0.572

log Kf

TABLE A2 Thermochemical Data of Selected Chemical Compounds Nitrogen, monatomic (N), ideal gas, molecular weight  14.0067 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

6.197 4.119 2.040 1.001

470.820 471.448 472.071 472.383

470.820 466.379 461.070 458.283

Infinite 243.611 120.419 95.753

153.300

0.

472.683

455.540

79.809

153.300 153.554 154.116 154.843 155.655

0.038 1.078 2.117 3.156 4.196

472.694 473.005 473.314 473.621 473.923

455.434 452.533 449.587 446.603 443.584

79.298 67.537 58.710 51.840 46.341

0 100 200 250

0. 20.786 20.786 20.786

0. 130.593 145.001 149.639

Infinite 171.780 155.201 153.642

298.15

20.786

153.300

300 350 400 450 500

20.788 20.786 20.786 20.786 20.786

153.429 156.633 159.408 161.857 164.047

615

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Nitrogen, monatomic (N), ideal gas, molecular weight  14.0067 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

600 700 800 900 1000

20.766 20.786 20.786 20.786 20.786

167.836 171.041 173.816 176.264 178.454

157.379 159.108 160.777 162.364 163.866

6.274 8.353 10.431 12.510 14.589

474.510 475.067 475.591 476.081 476.540

437.461 431.242 424.945 418.584 412.171

38.084 32.180 27.746 24.294 21.530

1100 1200 1300 1400 1500

20.786 20.786 20.788 20.786 20.786

180.436 182.244 183.908 185.448 186.882

165.284 166.623 167.889 169.089 170.228

16.667 18.746 20.824 22.903 24.982

476.970 477.374 477.756 478.118 478.462

405.713 399.217 392.688 386.131 379.548

19.266 17.377 15.778 14.407 13.217

1600 1700 1800 1900 2000

20.766 20.786 20.787 20.788 20.790

188.224 189.484 190.672 191.796 192.863

171.311 172.344 173.329 174.272 175.175

27.060 29.139 31.218 33.296 35.375

478.791 479.107 479.411 479.705 479.990

372.943 366.318 359.674 353.014 346.339

12.175 11.256 10.437 9.705 9.045

2100 2200 2300 2400 2500

20.793 20.797 20.804 20.813 20.826

193.877 194.844 195.769 196.655 197.504

176.042 176.874 177.676 178.448 179.194

37.454 39.534 41.614 43.695 45.777

480.266 480.536 480.799 481.057 481.311

339.650 332.947 326.233 319.507 312.770

8.448 7.905 7.409 6.954 6.535

2600 2700 2800 2900 3000

20.843 20.864 20.891 20.924 20.963

198.322 199.109 199.868 200.601 201.311

179.914 180.610 181.285 181.938 182.572

47.860 49.945 52.033 54.124 56.218

481.561 481.809 482.054 482.299 482.543

306.024 299.268 292.502 285.728 278.946

6.148 5.790 5.457 5.147 4.857

3100 3200 3300 3400 3500

21.010 21.064 21.126 21.197 21.277

202.000 202.667 203.317 203.948 204.564

183.188 183.786 184.368 184.935 185.487

58.317 60.420 62.530 64.646 66.769

482.789 483.036 483.286 483.540 483.799

272.155 265.357 258.550 251.736 244.915

4.586 4.332 4.093 3.867 3.655

3600 3700 3800 3900 4000

21.365 21.463 21.569 21.685 21.809

205.164 205.751 206.325 206.887 207.437

186.025 186.550 187.063 187.564 188.054

68.902 71.043 73.194 75.357 77.532

484.064 484.335 484.614 484.903 485.201

238.086 231.249 224.405 217.554 210.695

3.455 3.265 3.085 2.914 2.751

4100 4200 4300 4400 4500

21.941 22.082 22.231 22.388 22.551

207.977 208.508 209.029 209.542 210.047

188.534 189.003 189.463 189.913 190.355

79.719 81.920 84.136 86.367 88.614

485.510 485.830 486.164 486.510 486.871

203.829 196.955 190.073 183.183 176.285

2.597 2.449 2.309 2.175 2.046

4600 4700

22.722 22.899

210.544 211.035

190.788 191.214

90.877 93.158

487.247 487.638

169.379 162.465

1.923 1.806

616

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Nitrogen, monatomic (N), ideal gas, molecular weight  14.0067 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

4800 4900 5000

23.081 23.269 23.461

211.519 211.997 212.469

191.632 192.043 192.447

95.457 97.775 100.111

488.046 488.471 488.912

155.542 148.610 141.670

1.693 1.584 1.480

5100 5200 5300 5400 5500

23.658 23.858 24.061 24.266 24.474

212.935 213.397 213.853 214.305 214.752

192.844 193.235 193.619 193.998 194.371

102.467 104.843 107.238 109.655 112.092

489.372 489.849 490.345 490.860 491.394

134.721 127.762 120.794 113.817 106.829

1.380 1.283 1.190 1.101 1.015

5600 5700 5800 5900 6000

24.682 24.892 25.102 25.312 25.521

215.195 215.633 216.068 216.499 216.926

194.739 195.102 195.460 195.813 196.161

114.550 117.028 119.528 122.049 124.590

491.947 492.519 493.110 493.720 494.349

99.832 92.825 85.808 78.780 71.742

0.931 0.851 0.773 0.697 0.625

TABLE A2 Thermochemical Data of Selected Chemical Compounds Nitrogen oxide (NO), ideal gas, molecular weight  30.0061 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

0 100 200 250

0. 32.302 30.420 30.025

0. 177.031 198.747 205.488

Infinite 237.757 213.501 211.251

9.192 6.073 2.951 1.441

89.775 89.991 90.202 90.256

89.775 88.944 87.800 87.193

Infinite 46.460 22.931 18.218

298.15

29.845

210.758

210.758

0.

90.291

86.600

15.172

300 350 400 450 500

29.841 29.823 29.944 30.175 30.486

210.943 215.540 219.529 223.068 226.263

210.759 211.122 211.929 212.974 214.145

0.055 1.546 3.040 4.542 6.059

90.292 90.316 90.332 90.343 90.352

86.577 85.955 85.331 84.705 84.079

15.074 12.828 11.143 9.832 8.784

600 700 800 900 1000

31.238 32.028 32.767 33.422 33.987

231.886 236.761 241.087 244.985 248.536

216.646 219.179 221.652 224.031 226.307

9.144 12.307 15.548 18.858 22.229

90.366 90.381 90.398 90.417 90.437

82.822 81.564 80.303 79.041 77.775

7.210 6.086 5.243 4.587 4.063

617

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Nitrogen, monatomic (N), ideal gas, molecular weight  14.0067 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

1100 1200 1300 1400 1500

34.468 34.877 35.226 35.524 35.780

251.799 254.816 257.621 260.243 262.703

228.478 230.549 232.525 234.412 236.217

25.653 29.120 32.626 36.164 39.729

90.457 90.476 90.493 90.508 90.518

76.508 75.239 73.969 72.697 71.425

3.633 3.275 2.972 2.712 2.487

1600 1700 1800 1900 2000

36.002 36.195 36.364 36.514 36.647

265.019 267.208 269.282 271.252 273.128

237.945 239.603 241.195 242.725 244.199

43.319 46.929 50.557 54.201 57.859

90.525 90.526 90.522 90.511 90.494

70.151 68.878 67.605 66.332 65.060

2.290 2.116 1.962 1.824 1.699

2100 2200 2300 2400 2500

36.767 36.874 36.971 37.060 37.141

274.919 276.632 278.273 279.849 281.363

245.619 246.990 248.315 249.596 250.837

61.530 65.212 68.904 72.606 76.316

90.469 90.438 90.398 90.350 90.295

63.788 62.519 61.251 59.984 58.720

1.587 1.484 1.391 1.306 1.227

2600 2700 2800 2900 3000

37.216 37.285 37.350 37.410 37.466

282.822 284.227 285.585 286.896 288.165

252.039 253.205 254.338 255.438 256.508

80.034 83.759 87.491 91.229 94.973

90.231 90.160 90.081 89.994 89.899

57.458 56.199 54.943 53.689 52.439

1.154 1.087 1.025 0.967 0.913

3100 3200 3300 3400 3500

37.519 37.570 37.617 37.663 37.706

289.395 290.587 291.744 292.867 293.960

257.549 258.563 259.551 260.514 261.454

98.722 102.477 106.236 110.000 113.768

89.798 89.689 89.574 89.451 89.323

51.192 49.948 48.708 47.472 46.239

0.863 0.815 0.771 0.729 0.690

3600 3700 3800 3900 4000

37.747 37.787 37.825 37.862 37.898

295.022 296.057 297.065 298.048 299.008

262.372 263.269 264.145 265.902 265.840

117.541 121.318 125.098 128.883 132.671

89.189 89.049 88.903 88.752 88.596

45.010 43.784 42.563 41.346 40.132

0.653 0.618 0.585 0.554 0.524

4100 4200 4300 4400 4500

37.933 37.966 37.999 38.031 38.062

299.944 300.858 301.752 302.626 303.481

266.660 267.464 268.251 269.022 269.778

136.462 140.257 144.056 147.857 151.662

88.434 88.268 88.097 87.922 87.741

38.922 37.717 36.515 35.318 34.124

0.496 0.469 0.444 0.419 0.396

4600 4700 4800 4900 5000

38.092 38.122 38.151 38.180 38.208

304.318 305.137 305.940 306.727 307.499

270.520 271.248 271.962 272.664 273.353

155.469 159.280 163.094 166.910 170.730

87.556 87.366 87.171 86.970 86.765

32.934 31.749 30.568 29.391 28.218

0.374 0.353 0.333 0.313 0.295

5100 5200 5300

38.235 38.262 38.289

308.256 308.998 309.728

274.030 274.695 275.349

174.552 178.377 182.204

86.553 86.336 86.112

27.049 25.884 24.724

0.277 0.260 0.244

618

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Nitrogen oxide (NO), ideal gas, molecular weight  30.0061 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p S° (J/K/mol) (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

5400 5500

38.316 38.342

310.443 311.147

275.993 276.625

186.034 189.867

85.881 85.644

23.568 22.416

0.228 0.213

5600 5700 5800 5900 6000

38.367 38.393 38.418 38.443 38.468

311.838 312.517 313.185 313.842 314.488

277.248 277.861 278.464 279.058 279.643

193.703 197.541 201.381 205.224 209.070

85.399 85.146 84.884 84.613 84.331

21.269 20.125 18.987 17.853 16.724

0.198 0.184 0.171 0.158 0.146

TABLE A2 Thermochemical Data of Selected Chemical Compounds Nitrogen oxide, ion (NO), ideal gas, molecular weight  30.00555 Enthalpy reference temperature  Tr  298.15 K [G°  H°(Tr)]/T (J/K/mol)

C°p (J/K/mol)

S° (J/K/mol)

0 100 200 250

0. 29.104 29.107 29.111

0. 166.421 186.595 193.091

Infinite 224.100 200.882 198.698

298.15

29.123

198.219

300 350 400 450 500

29.124 29.163 29.244 29.378 29.568

198.399 202.891 206.790 210.242 213.346

600 700 800 900 1000

30.089 30.728 31.403 32.059 32.666

1100 1200 1300 1400 1500 1600 1700 1800

T (K)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

8.670 5.768 2.857 1.402

983.995

198.219

0.

990.185

983.978

172.389

198.219 198.574 199.363 200.383 201.527

0.054 1.511 2.971 4.436 5.910

990.224 991.253 992.275 993.288 994.293

983.939 982.811 981.535 980.131 978.615

171.319 146.676 128.175 113.771 102.235

218.781 223.466 227.613 231.350 234.760

203.962 206.421 208.816 211.115 213.312

8.891 11.931 15.038 18.211 21.448

996.282 998.252 1000.214 1002.175 1004.139

975.293 971.639 967.703 963.521 959.121

84.907 72.505 63.185 55.921 50.099

33.213 33.697 34.124 34.497 34.825

237.899 240.810 243.525 246.067 248.459

215.406 217.403 219.309 221.131 222.874

24.742 28.088 31.480 34.911 38.378

1006.109 1008.085 1010.066 1012.053 1014.043

954.524 949.747 944.805 939.711 934.474

45.327 41.341 37.963 35.061 32.541

35.112 35.365 35.588

250.716 252.852 254.880

224.544 226.147 227.687

41.875 45.399 48.947

1016.036 1018.030 1020.024

929.104 923.610 917.998

30.332 28.379 26.640

619

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Nitrogen oxide, ion (NO), ideal gas, molecular weight  30.00555 Enthalpy reference temperature  Tr  298.15 K [G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

T (K)

C°p (J/K/mol)

S° (J/K/mol)

1900 2000

35.786 35.963

256.810 258.650

229.170 230.598

52.516 56.103

1022.017 1024.008

912.276 906.449

25.080 23.674

2100 2200 2300 2400 2500

36.121 36.263 36.391 36.507 36.612

260.408 262.092 263.707 265.258 266.750

231.976 233.307 234.594 235.839 237.046

59.708 63.327 66.960 70.605 74.261

1025.995 1027.979 1029.959 1031.933 1033.902

900.522 894.501 888.389 882.191 875.911

22.399 21.238 20.176 19.200 18.301

2600 2700 2800 2900 3000

36.709 36.798 36.880 36.955 37.025

268.188 269.575 270.915 272.211 273.465

238.216 239.352 240.456 241.529 242.572

77.927 81.602 85.286 88.978 92.677

1035.865 1037.823 1039.774 1041.720 1043.659

869.553 863.119 856.613 850.037 843.395

17.470 16.698 15.980 15.311 14.685

3100 3200 3300 3400 3500

37.091 37.152 37.209 37.263 37.313

274.680 275.858 277.003 278.114 279.195

243.589 244.579 245.544 246.486 247.405

96.383 100.095 103.813 107.537 111.266

1045.593 1047.520 1049.442 1051.358 1053.269

836.687 829.917 823.087 816.200 809.255

14.098 13.547 13.028 12.539 12.077

3600 3700 3800 3900 4000

37.361 37.407 37.450 37.491 37.531

280.247 281.271 282.269 283.243 284.192

248.303 249.180 250.038 250.877 251.698

114.999 118.738 122.481 126.228 129.979

1055.174 1057.074 1058.969 1060.860 1062.745

802.257 795.205 788.102 780.950 773.748

11.640 11.226 10.833 10.460 10.104

4100 4200 4300 4400 4500

37.569 37.605 37.640 37.673 37.705

285.120 286.025 286.911 287.776 288.623

252.502 253.289 254.061 254.817 255.559

133.734 137.492 141.255 145.020 148.789

1064.626 1066.502 1068.374 1070.241 1072.103

766.500 759.206 751.867 744.485 737.060

9.765 9.442 9.133 8.838 8.556

4600 4700 4800 4900 5000

37.737 37.767 37.796 37.825 37.853

289.452 290.264 291.060 291.839 292.604

256.287 257.001 257.702 258.391 259.068

152.561 156.337 160.115 163.896 167.680

1073.961 1075.814 1077.662 1079.505 1081.342

729.594 722.088 714.542 706.959 699.337

8.285 8.025 7.776 7.536 7.306

5100 5200 5300 5400 5500

37.880 37.906 37.932 37.957 37.982

293.354 294.089 294.812 295.521 296.218

259.733 260.386 260.029 261.661 262.283

171.466 175.256 179.048 182.842 186.639

1083.174 1084.999 1086.819 1088.631 1090.436

691.679 683.984 676.255 668.492 660.694

7.084 6.871 6.665 6.466 6.275

5600 5700 5800 5900 6000

38.006 38.030 38.053 38.076 38.098

296.902 297.575 298.237 298.887 299.528

262.895 263.498 264.091 264.675 265.251

190.438 194.240 198.044 201.851 205.660

1092.234 1094.023 1095.803 1097.574 1099.334

652.865 645.003 637.110 629.186 621.233

6.090 5.911 5.738 5.570 5.408

log Kf

620

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Nitrogen dioxide (NO2), ideal gas, molecular weight  46.0055 Enthalpy reference temperature  Tr  298.15 K [G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

C°p (J/K/mol)

S° (J/K/mol)

0 100 200 250

0. 33.276 34.385 35.593

0. 202.563 225.852 233.649

Infinite 271.168 243.325 240.634

10.186 6.861 3.495 1.746

35.927 34.898 33.897 33.460

35.927 39.963 45.422 48.355

Infinite 20.874 11.863 10.103

298.15

36.974

240.034

240.034

0.

33.095

51.258

8.980

300 350 400 450 500

37.029 38.583 40.171 41.728 43.206

240.262 246.086 251.342 256.164 260.638

240.034 240.491 241.524 242.886 244.440

0.068 1.958 3.927 5.975 8.099

33.083 32.768 32.512 32.310 32.154

51.371 54.445 57.560 60.703 63.867

8.944 8.125 7.517 7.046 6.672

600 700 800 900 1000

45.834 47.986 49.708 51.076 52.166

268.755 275.988 282.512 288.449 293.889

247.830 251.345 254.840 258.250 261.545

12.555 17.250 22.138 27.179 32.344

31.959 31.878 31.874 31.923 32.005

70.230 76.616 83.008 89.397 95.779

6.114 5.717 5.420 5.188 5.003

1100 1200 1300 1400 1500

53.041 53.748 54.326 54.803 55.200

298.903 303.550 307.876 311.920 315.715

264.717 267.761 270.683 273.485 276.175

37.605 42.946 48.351 53.808 59.309

32.109 32.226 32.351 32.478 32.603

102.152 108.514 114.867 121.209 127.543

4.851 4.724 4.615 4.522 4.441

1600 1700 1800 1900 2000

55.533 55.815 56.055 56.262 56.441

319.288 322.663 325.861 328.897 331.788

278.759 281.244 283.634 285.937 288.158

64.846 70.414 76.007 81.624 87.259

32.724 32.837 32.940 33.032 33.111

133.868 140.186 146.497 152.804 159.106

4.370 4.307 4.251 4.201 4.155

2100 2200 2300 2400 2500

56.596 56.732 56.852 56.958 57.052

334.545 337.181 339.706 342.128 344.455

290.302 292.373 294.377 296.316 298.196

92.911 98.577 104.257 109.947 115.648

33.175 33.223 33.255 33.270 33.268

165.404 171.700 177.993 184.285 190.577

4.114 4.077 4.042 4.011 3.982

2600 2700 2800 2900 3000

57.136 57.211 57.278 57.339 57.394

346.694 348.852 350.934 352.945 354.889

300.018 301.787 303.505 305.176 306.800

121.357 127.075 132.799 138.530 144.267

33.248 33.210 33.155 33.082 32.992

196.870 203.164 209.460 215.757 222.058

3.955 3.930 3.908 3.886 3.866

3100 3200 3300 3400 3500

57.444 57.490 57.531 57.569 57.604

356.772 358.597 360.366 362.084 363.754

308.382 309.923 311.425 312.890 314.319

150.009 155.756 161.507 167.262 173.020

32.885 32.761 32.622 32.467 32.297

228.363 234.670 240.981 247.298 253.618

3.848 3.831 3.814 3.799 3.785

3600 3700 3800 3900 4000

57.636 57.666 57.693 57.719 57.742

365.377 366.957 368.495 369.994 371.455

315.715 317.079 318.412 319.715 320.991

178.783 184.548 190.316 196.086 201.859

32.113 31.914 31.701 31.475 31.236

259.945 266.276 272.613 278.956 285.305

3.772 3.759 3.747 3.736 3.726

T (K)

log Kf

621

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Nitrogen dioxide (NO2), ideal gas, molecular weight  46.0055 Enthalpy reference temperature  Tr  298.15 K [G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

T (K)

C°p (J/K/mol)

S° (J/K/mol)

4100 4200 4300 4400 4500

57.764 57.784 57.803 57.821 57.837

372.881 374.274 375.634 376.963 378.262

322.239 323.461 324.659 325.833 326.983

207.635 213.412 219.191 224.973 230.756

30.985 30.720 30.444 30.155 29.854

291.659 298.020 304.388 310.762 317.142

3.716 3.706 3.698 3.689 3.681

4600 4700 4800 4900 5000

57.853 57.867 57.881 57.894 57.906

379.534 380.778 381.996 383.190 384.360

328.112 329.219 330.306 331.373 332.421

236.540 242.326 248.114 253.902 259.692

29.540 29.214 28.875 28.523 28.158

323.530 329.925 336.326 342.736 349.152

3.674 3.667 3.660 3.654 3.648

5100 5200 5300 5400 5500

57.917 57.928 57.938 57.948 57.957

385.507 386.631 387.735 388.818 389.881

333.451 334.463 335.458 336.436 337.398

265.483 271.276 277.069 282.863 288.658

27.778 27.384 26.974 26.548 26.106

355.576 362.006 368.446 374.892 381.347

3.642 3.636 3.631 3.626 3.622

5600 5700 5800 5900 6000

57.965 57.973 57.981 57.988 57.995

390.926 391.952 392.960 393.951 394.926

338.344 339.276 340.193 341.096 341.985

294.455 300.251 306.049 311.848 317.647

25.646 25.167 24.669 24.150 23.608

387.811 394.281 400.762 407.249 413.748

3.617 3.613 3.609 3.606 3.602

log Kf

TABLE A2 Thermochemical Data of Selected Chemical Compounds Nitrogen (N2), ideal gas-reference state, molecular weight  28.0134 Enthalpy reference temperature  Tr  298.15 K [G°  H°(Tr)]/T (J/K/mol)

C°p (J/K/mol)

S° (J/K/mol)

0 100 200 250

0. 29.104 29.107 29.111

0. 159.811 179.985 186.481

Infinite 217.490 194.272 192.088

298.15

29.124

191.609

300 350 400 450 500

29.125 29.165 29.249 29.387 29.580

191.789 196.281 200.181 203.633 206.739

600 700 800

30.110 30.754 31.433

212.176 216.866 221.017

T (K)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

8.670 5.768 2.857 1.402

0. 0. 0. 0.

0. 0. 0. 0.

0. 0. 0. 0.

191.609

0.

0.

0.

0.

191.610 191.964 192.753 193.774 194.917

0.054 1.511 2.971 4.437 5.911

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

197.353 199.813 202.209

8.894 11.937 15.046

0. 0. 0.

0. 0. 0.

0. 0. 0.

622

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Nitrogen (N2), ideal gas-reference state, molecular weight  28.0134 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

900 1000

32.090 32.697

224.757 228.170

204.510 206.708

18.223 21.463

0. 0.

0. 0.

0. 0.

1100 1200 1300 1400 1500

33.241 33.723 34.147 34.518 34.843

231.313 234.226 236.943 239.487 241.880

208.804 210.802 212.710 214.533 216.277

24.760 28.109 31.503 34.936 38.405

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

1600 1700 1800 1900 2000

35.128 35.378 35.600 35.796 35.971

244.138 246.275 248.304 250.234 252.074

217.948 219.552 221.094 222.577 224.006

41.904 45.429 48.978 52.548 56.137

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

2100 2200 2300 2400 2500

36.126 36.268 36.395 36.511 36.616

253.833 255.517 257.132 258.684 260.176

225.385 226.717 228.004 229.250 230.458

59.742 63.361 66.995 70.640 74.296

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

2600 2700 2800 2900 3000

36.713 36.801 36.883 36.959 37.030

261.614 263.001 264.341 265.637 266.891

231.629 232.765 233.869 234.942 235.986

77.963 81.639 85.323 89.015 92.715

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

3100 3200 3300 3400 3500

37.096 37.158 37.216 37.271 37.323

268.106 269.285 270.429 271.541 272.622

237.003 237.993 238.959 239.901 240.821

96.421 100.134 103.852 107.577 111.306

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

3600 3700 3800 3900 4000

37.373 37.420 37.465 37.508 37.550

273.675 274.699 275.698 276.671 277.622

241.719 242.596 243.454 244.294 245.115

115.041 118.781 122.525 126.274 130.027

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

4100 4200 4300 4400 4500

37.590 37.629 37.666 37.702 37.738

278.549 279.456 280.341 281.208 282.056

245.919 246.707 247.479 248.236 248.978

133.784 137.545 141.309 145.078 148.850

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

4600 4700 4800 4900 5000

37.773 37.808 37.843 37.878 37.912

282.885 283.698 284.494 285.275 286.041

249.706 250.420 251.122 251.811 252.488

152.625 156.405 160.187 163.973 167.763

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

5100 5200 5300 5400 5500

37.947 37.981 38.013 38.046 38.080

286.792 287.529 288.253 288.964 289.662

253.153 253.807 254.451 255.083 255.705

171.556 175.352 179.152 182.955 186.761

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

log Kf

623

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Nitrogen (N2), ideal gas-reference state, molecular weight  28.0134 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

5600 5700 5800 5900 6000

38.116 38.154 38.193 38.234 38.276

290.348 291.023 291.687 292.341 292.984

256.318 256.921 257.515 258.099 258.675

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol) 190.571 194.384 198.201 202.023 205.848

0. 0. 0. 0. 0.

ΔfG° (kJ/mol) 0. 0. 0. 0. 0.

log Kf 0. 0. 0. 0. 0.

TABLE A2 Thermochemical Data of Selected Chemical Compounds Oxygen, monatomic (O), ideal gas, molecular weight  15.9994 Enthalpy reference temperature  Tr  298.15 K C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

0 100 200 250

0. 23.703 22.734 22.246

0. 135.947 152.153 157.170

Infinite 181.131 163.085 161.421

298.15

21.911

161.058

300 350 400 450 500

21.901 21.657 21.482 21.354 21.257

161.194 164.551 167.430 169.953 172.197

600 700 800 900 1000

21.124 21.040 20.984 20.944 20.915

1100 1200 1300 1400 1500

T (K)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

6.725 4.518 2.186 1.063

246.790 247.544 248.421 248.816

246.790 242.615 237.339 234.522

Infinite 126.729 61.986 49.001

161.058

0.

249.173

231.736

40.599

161.059 161.324 161.912 162.668 163.511

0.041 1.129 2.207 3.278 4.343

249.187 249.537 249.868 250.180 250.474

231.628 228.673 225.670 222.626 219.549

40.330 34.128 29.469 25.842 22.936

176.060 179.310 182.116 184.585 186.790

165.291 167.067 168.777 170.399 171.930

6.462 8.570 10.671 12.767 14.860

251.013 251.494 251.926 252.320 252.682

213.312 206.990 200.602 194.163 187.681

18.570 15.446 13.098 11.269 9.803

20.893 20.877 20.864 20.853 20.845

188.782 190.599 192.270 193.816 195.254

173.373 174.734 176.019 177.236 178.390

16.950 19.039 21.126 23.212 25.296

253.018 253.332 253.627 253.906 254.171

181.165 174.619 168.047 161.453 154.840

8.603 7.601 6.752 6.024 5.392

1600 1700 1800 1900 2000

20.838 20.833 20.830 20.827 20.826

196.599 197.862 199.053 200.179 201.247

179.486 180.530 181.527 182.479 183.391

27.381 29.464 31.547 33.630 35.713

254.421 254.659 254.884 255.097 255.299

148.210 141.564 134.905 128.234 121.552

4.839 4.350 3.915 3.525 3.175

2100 2200

20.827 20.830

202.263 203.232

184.266 185.106

37.796 39.878

255.488 255.667

114.860 108.159

2.857 2.568

624

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Oxygen, monatomic (O), ideal gas, molecular weight  15.9994 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

2300 2400 2500

20.835 20.841 20.851

204.158 205.045 205.896

185.914 186.693 187.444

41.962 44.045 46.130

255.835 255.992 256.139

101.450 94.734 88.012

2.304 2.062 1.839

2600 2700 2800 2900 3000

20.862 20.877 20.894 20.914 20.937

206.714 207.502 208.261 208.995 209.704

188.170 188.871 189.550 190.208 190.846

48.216 50.303 52.391 54.481 56.574

256.277 256.405 256.525 256.637 256.741

81.284 74.551 67.814 61.072 54.327

1.633 1.442 1.265 1.100 0.946

3100 3200 3300 3400 3500

20.963 20.991 21.022 21.056 21.092

210.391 211.057 211.704 212.332 212.943

191.466 192.068 192.653 193.223 193.777

58.669 60.767 62.867 64.971 67.079

256.838 256.929 257.014 257.094 257.169

47.578 40.826 34.071 27.315 20.555

0.802 0.666 0.539 0.420 0.307

3600 3700 3800 3900 4000

21.130 21.170 21.213 21.257 21.302

213.537 214.117 214.682 215.234 215.772

194.318 194.845 195.360 195.862 196.353

69.190 71.305 73.424 75.547 77.675

257.241 257.309 257.373 257.436 257.496

13.794 7.030 0.265 6.501 13.270

0.200 0.099 0.004 0.087 0.173

4100 4200 4300 4400 4500

21.349 21.397 21.445 21.495 21.545

216.299 216.814 217.318 217.812 218.295

196.834 197.303 197.763 198.213 198.654

79.808 81.945 84.087 86.234 88.386

257.554 257.611 257.666 257.720 257.773

20.010 26.811 33.583 40.358 47.133

0.255 0.333 0.408 0.479 0.547

4600 4700 4800 4900 5000

21.596 21.647 21.697 21.748 21.799

218.769 219.234 219.690 220.138 220.578

199.086 199.510 199.925 200.333 200.734

90.543 92.705 94.872 97.045 99.222

257.825 257.876 257.926 257.974 258.021

53.909 60.687 67.465 74.244 81.025

0.612 0.674 0.734 0.791 0.846

5100 5200 5300 5400 5500

21.849 21.899 21.949 21.997 22.045

221.010 221.435 221.853 222.264 222.668

201.127 201.514 201.893 202.267 202.634

101.405 103.592 105.784 107.982 110.184

258.066 258.110 258.150 258.189 258.224

87.806 94.589 101.371 108.155 114.940

0.899 0.950 0.999 1.046 1.092

5600 5700 5800 5900 6000

22.093 22.139 22.184 22.229 22.273

223.065 223.457 223.842 224.222 224.596

202.995 203.351 203.701 204.046 204.385

112.391 114.602 116.818 119.039 121.264

258.255 258.282 258.304 258.321 258.332

121.725 128.510 135.296 142.083 148.869

1.135 1.178 1.218 1.258 1.296

log Kf

625

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Sulfur Monoxide (SO), ideal gas, molecular weight  48.0594 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

0 100 200 250

0. 29.106 29.269 29.634

0. 189.916 210.114 216.680

Infinite 248.172 224.668 222.437

8.733 5.826 2.911 1.439

5.028 5.793 5.610 5.333

5.028 3.281 12.337 16.793

Infinite 1.714 3.222 3.509

298.15

30.173

221.944

221.944

0.

5.007

21.026

3.684

300 350 400 450 500

30.197 30.867 31.560 32.221 32.826

222.130 226.835 231.002 234.758 238.184

221.944 222.314 223.144 224.230 225.456

0.056 1.582 3.143 4.738 6.364

4.994 4.616 1.998 0.909 0.238

21.187 25.521 29.711 33.619 37.391

3.689 3.809 3.880 3.902 3.906

600 700 800 900 1000

33.838 34.612 35.206 35.672 36.053

244.263 249.540 254.202 258.377 262.155

228.097 230.791 233.432 235.976 238.408

9.699 13.124 16.616 20.161 23.748

2.067 3.617 5.005 59.419 59.420

44.643 51.614 58.374 63.886 64.382

3.887 3.851 3.811 3.708 3.363

1100 1200 1300 1400 1500

36.379 36.672 36.946 37.210 37.469

265.607 268.785 271.731 274.479 277.055

240.726 242.933 245.037 247.043 248.959

27.369 31.022 34.703 38.411 42.145

59.424 59.432 59.446 59.462 59.482

64.878 65.374 65.869 66.362 66.854

3.081 2.846 2.647 2.476 2.328

1600 1700 1800 1900 2000

37.725 37.980 38.232 38.482 38.727

279.482 281.776 283.954 286.028 288.008

250.791 252.547 254.232 255.851 257.410

45.905 49.690 53.501 57.336 61.197

59.505 59.528 59.552 59.575 59.597

67.345 67.834 68.323 68.809 69.294

2.199 2.084 1.983 1.892 1.810

2100 2200 2300 2400 2500

38.967 39.200 39.425 39.641 39.847

289.904 291.722 293.469 295.152 296.774

258.912 260.363 261.764 263.121 264.434

65.082 68.990 72.921 76.875 80.849

59.618 59.636 59.652 59.666 59.678

69.779 70.262 70.745 71.227 71.708

1.736 1.668 1.607 1.550 1.498

2600 2700 2800 2900 3000

40.043 40.229 40.404 40.568 40.721

298.341 299.856 301.322 302.742 304.120

265.709 266.945 268.147 269.316 270.453

84.844 88.857 92.889 96.938 101.002

59.687 59.694 59.699 59.702 59.705

72.189 72.670 73.150 73.631 74.111

1.450 1.406 1.365 1.326 1.290

3100 3200 3300 3400 3500

40.864 40.996 41.119 41.232 41.336

305.458 306.757 308.021 309.250 310.447

271.561 272.640 273.693 274.721 275.725

105.082 109.175 113.281 117.398 121.527

59.707 59.708 59.710 59.712 59.716

74.591 75.072 75.552 76.032 76.512

1.257 1.225 1.196 1.168 1.142

3600 3700 3800 3900 4000

41.432 41.520 41.601 41.676 41.745

311.613 312.749 313.857 314.939 315.995

276.706 277.665 278.603 279.520 280.419

125.665 129.813 133.969 138.133 142.304

59.722 59.730 59.741 59.755 59.774

76.991 77.471 77.950 78.429 78.908

1.117 1.094 1.071 1.050 1.030

T (K)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

626

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Sulfur Monoxide (SO), ideal gas, molecular weight  48.0594 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

4100 4200 4300 4400 4500

41.810 41.871 41.929 41.986 42.042

317.027 318.035 319.021 319.985 320.930

281.299 282.162 283.008 283.837 284.651

146.482 150.666 154.856 159.051 163.253

59.796 59.824 59.857 59.896 59.941

79.386 79.863 80.340 80.816 81.291

1.011 0.993 0.976 0.959 0.944

4600 4700 4800 4900 5000

42.098 42.156 42.217 42.282 42.352

321.854 322.760 323.648 324.520 325.374

285.450 286.234 287.004 287.761 288.505

167.460 171.673 175.891 180.116 184.348

59.992 60.049 60.113 60.183 60.259

81.765 82.238 82.709 83.179 83.648

0.928 0.914 0.900 0.887 0.874

5100 5200 5300 5400 5500

42.429 42.514 42.608 42.712 42.829

326.214 327.039 327.849 328.647 329.431

289.236 289.955 290.663 291.359 292.044

188.587 192.834 197.090 201.356 205.633

60.342 60.430 60.524 60.623 60.726

84.114 84.580 85.043 85.505 85.965

0.862 0.850 0.838 0.827 0.816

5600 5700 5800 5900 6000

42.959 43.104 43.265 43.444 43.642

330.204 330.966 331.717 332.458 333.190

292.718 293.383 294.037 294.682 295.318

209.922 214.225 218.543 222.879 227.233

60.832 60.941 61.051 61.162 61.271

86.423 86.879 87.333 87.786 88.235

0.806 0.796 0.787 0.777 0.768

log Kf

TABLE A2 Thermochemical Data of Selected Chemical Compounds Titanium oxide (TiO), ideal gas, molecular weight  63.8794 Enthalpy reference temperature  Tr  298.15 K C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

0 100 200 250

0. 33.880 31.771 31.906

0. 198.046 220.717 227.809

Infinite 262.006 236.408 234.005

298.15

32.476

233.474

300 350 400 450 500

32.502 33.263 34.021 34.702 35.285

233.675 238.741 243.233 247.280 250.968

600 700 800

36.176 36.786 37.212

257.485 263.110 268.051

T (K)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

9.630 6.396 3.138 1.549

53.934 55.155 55.040 54.737

53.934 44.842 34.491 29.387

Infinite 23.423 9.008 6.140

233.474

0.

54.392

24.535

4.298

233.474 233.873 234.767 235.937 237.258

0.060 1.704 3.386 5.105 6.855

54.378 53.997 53.605 53.208 52.804

24.350 19.375 14.456 9.586 4.760

4.240 2.892 1.888 1.113 0.497

240.100 242.995 245.824

10.431 14.081 17.782

51.975 51.109 50.217

4.771 14.161 23.425

0.415 1.057 1.529

627

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Titanium oxide (TiO), ideal gas, molecular weight  63.8794 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

900 1000

37.518 37.745

272.453 276.418

248.543 251.136

21.519 25.283

49.261 48.172

32.573 41.609

1.891 2.173

1100 1200 1300 1400 1500

37.918 38.052 38.158 38.245 38.319

280.024 283.330 286.380 289.211 291.852

253.600 255.942 258.168 260.285 262.302

29.066 32.865 36.676 40.496 44.324

46.886 41.416 40.455 39.409 38.267

50.527 59.198 67.545 75.814 84.005

2.399 2.577 2.714 2.829 2.925

1600 1700 1800 1900 2000

38.386 38.450 38.516 38.588 38.669

294.327 296.656 298.856 300.940 302.922

264.227 266.067 267.828 269.517 271.138

48.160 52.001 55.850 59.705 63.568

37.014 35.637 34.124 32.461 15.951

92.116 100.145 108.089 115.945 123.256

3.007 3.077 3.137 3.188 3.219

2100 2200 2300 2400 2500

38.762 38.869 38.992 39.132 39.290

304.810 306.616 308.346 310.009 311.609

272.697 274.198 275.645 277.042 278.393

67.439 71.321 75.213 79.120 83.040

13.206 10.459 7.713 4.969 2.229

130.149 136.911 143.548 150.066 156.469

3.237 3.251 3.260 3.266 3.269

2600 2700 2800 2900 3000

39.466 39.661 39.874 40.105 40.352

313.154 314.647 316.093 317.496 318.860

279.701 280.967 282.196 283.389 284.549

86.978 90.934 94.911 98.910 102.932

0.505 3.232 5.948 8.651 11.341

162.763 168.952 175.040 181.032 186.930

3.270 3.269 3.265 3.261 3.255

3100 3200 3300 3400 3500

40.614 40.890 41.179 41.478 41.787

320.187 321.481 322.744 323.978 325.184

285.677 286.776 287.847 288.892 289.911

106.981 111.056 115.159 119.292 123.455

14.014 16.670 19.306 21.921 24.513

192.738 198.462 204.102 209.662 215.146

3.248 3.240 3.231 3.221 3.211

3600 3700 3800 3900 4000

42.103 42.425 42.751 43.080 43.409

326.366 327.524 328.660 329.774 330.869

290.908 291.882 292.835 293.768 294.681

127.649 131.876 136.135 140.426 144.751

27.082 438.725 440.010 441.324 442.666

220.557 218.108 212.127 206.113 200.065

3.200 3.079 2.916 2.761 2.613

4100 4200 4300 4400 4500

43.738 44.064 44.386 44.703 45.015

331.945 333.003 334.044 335.068 336.076

295.577 296.456 297.318 298.164 298.996

149.108 153.498 157.921 162.375 166.861

444.034 445.427 446.843 448.281 449.739

193.983 187.868 181.719 175.537 169.322

2.471 2.336 2.207 2.084 1.965

4600 4700 4800 4900 5000

45.319 45.614 45.901 46.178 46.444

337.068 338.046 339.010 339.959 340.895

299.812 300.616 301.405 302.183 302.947

171.378 175.925 180.500 185.104 189.736

451.216 452.711 454.222 455.750 457.293

163.074 156.794 150.482 144.138 137.763

1.852 1.743 1.638 1.537 1.439

5100 5200 5300 5400 5500

46.699 46.943 47.174 47.394 47.602

341.817 342.726 343.622 344.506 345.378

303.701 304.442 305.173 305.893 306.603

194.393 199.075 203.781 208.509 213.259

458.850 460.421 462.005 463.602 465.211

131.357 124.920 118.453 111.956 105.430

1.345 1.255 1.167 1.083 1.001

log Kf

628

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Titanium oxide (TiO), ideal gas, molecular weight  63.8794 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

5500 5600 5700 5800 5900 6000

47.602 47.797 47.980 48.150 48.309 48.455

345.378 346.237 347.085 347.921 348.745 349.558

306.603 307.303 307.994 308.675 309.347 310.011

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol) 213.259 218.029 222.818 227.625 232.448 237.286

ΔfG° (kJ/mol)

465.211 466.833 468.467 470.113 471.772 473.442

105.430 98.873 92.288 85.674 79.032 72.361

log Kf 1.001 0.922 0.846 0.772 0.700 0.630

TABLE A2 Thermochemical Data of Selected Chemical Compounds Oxygen (O2), ideal gas-reference state, molecular weight  31.9988 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

log Kf

0 100 200 250

0. 29.106 29.126 29.201

0. 173.307 193.485 199.990

Infinite 231.094 207.823 205.630

8.683 5.779 2.868 1.410

0. 0. 0. 0.

0. 0. 0. 0.

0. 0. 0. 0.

298.15

29.376

205.147

205.147

0.

0.

0.

0.

300 350 400 450 500

29.385 29.694 30.106 30.584 31.091

205.329 209.880 213.871 217.445 220.693

205.148 205.506 206.308 207.350 208.524

0.054 1.531 3.025 4.543 6.084

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

600 700 800 900 1000

32.090 32.981 33.733 34.355 34.870

226.451 231.466 235.921 239.931 243.578

211.044 213.611 216.126 218.552 220.875

9.244 12.499 15.835 19.241 22.703

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

1100 1200 1300 1400 1500

35.300 35.667 35.988 36.277 36.544

246.922 250.010 252.878 255.556 258.068

223.093 225.209 227.229 229.158 231.002

26.212 29.761 33.344 36.957 40.599

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

1600 1700 1800 1900 2000

36.796 37.040 37.277 37.510 37.741

260.434 262.672 264.796 266.818 268.748

232.768 234.462 236.089 237.653 239.160

44.266 47.958 51.673 55.413 59.175

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

2100 2200 2300

37.969 38.195 38.419

270.595 272.366 274.069

240.613 242.017 243.374

62.961 66.769 70.600

0. 0. 0.

0. 0. 0.

0. 0. 0.

T (K)

629

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Oxygen (O2), ideal gas-reference state, molecular weight  31.9988 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

log Kf

2400 2500

38.639 38.856

275.709 277.290

244.687 245.959

74.453 78.328

0. 0.

0. 0.

0. 0.

2600 2700 2800 2900 3000

39.068 39.276 39.478 39.674 39.864

278.819 280.297 281.729 283.118 284.466

247.194 248.393 249.558 250.691 251.795

82.224 86.141 90.079 94.036 98.013

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

3100 3200 3300 3400 3500

40.048 40.225 40.395 40.559 40.716

285.776 287.050 288.291 289.499 290.677

252.870 253.918 254.941 255.940 256.916

102.009 106.023 110.054 114.102 118.165

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

3600 3700 3800 3900 4000

40.868 41.013 41.154 41.289 41.421

291.826 292.948 294.044 295.115 296.162

257.870 258.802 259.716 260.610 261.485

122.245 126.339 130.447 134.569 138.705

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

4100 4200 4300 4400 4500

41.549 41.674 41.798 41.920 42.042

297.186 298.189 299.171 300.133 301.076

262.344 263.185 264.011 264.821 265.616

142.854 147.015 151.188 155.374 159.572

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

4600 4700 4800 4900 5000

42.164 42.287 42.413 42.542 42.675

302.002 302.910 303.801 304.677 305.538

266.397 267.164 267.918 268.660 269.389

163.783 168.005 172.240 176.488 180.749

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

5100 5200 5300 5400 5500

42.813 42.956 43.105 43.262 43.426

306.385 307.217 308.037 308.844 309.639

270.106 270.811 271.506 272.190 272.864

185.023 189.311 193.614 197.933 202.267

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

5600 5700 5800 5900 6000

43.599 43.781 43.973 44.175 44.387

310.424 311.197 311.960 312.713 313.457

273.527 274.181 274.826 275.462 276.089

206.618 210.987 215.375 219.782 224.210

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

630

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Sulfur dioxide (SO2), ideal gas, molecular weight  64.0588 Enthalpy reference temperature  Tr  298.15 K C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

0 100 200

0. 33.526 36.372

0. 209.025 233.033

298.15

39.878

300 400 500

39.945 43.493 46.576

600 700 800 900 1000

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

Infinite 281.199 251.714

10.552 7.217 3.736

294.299 294.559 295.631

294.299 296.878 298.813

Infinite 155.073 78.042

248.212

248.212

0.000

296.842

300.125

52.581

248.459 260.448 270.495

248.213 249.824 252.979

0.074 4.250 8.758

296.865 300.257 302.736

300.145 300.971 300.871

52.260 39.303 31.432

49.049 50.961 52.434 53.580 54.484

279.214 286.924 293.829 300.073 305.767

256.641 260.427 264.178 267.825 271.339

13.544 18.548 23.721 29.023 34.428

304.694 306.291 307.667 362.026 361.940

300.305 299.444 298.370 296.051 288.725

26.144 22.345 19.482 17.182 15.081

1100 1200 1300 1400 1500

55.204 55.794 56.279 56.689 57.036

310.995 315.824 320.310 324.496 328.419

274.710 277.937 281.026 283.983 286.816

39.914 45.464 51.069 56.718 62.404

361.835 361.720 361.601 361.484 361.372

281.109 274.102 266.806 259.518 252.239

13.363 11.931 10.720 9.683 8.784

1600 1700 1800 1900 2000

57.338 57.601 57.831 58.040 58.229

332.110 335.594 338.893 342.026 345.007

289.533 292.141 294.647 297.059 299.383

68.123 73.870 79.642 85.436 91.250

361.268 361.176 361.096 361.031 360.981

244.967 237.701 230.440 223.183 215.929

7.997 7.304 6.687 6.136 5.639

2100 2200 2300 2400 2500

58.400 58.555 58.702 58.840 58.965

347.853 350.573 353.179 355.680 358.085

301.624 303.787 305.878 307.902 309.861

97.081 102.929 108.792 114.669 120.559

360.948 360.931 360.930 360.947 360.980

208.678 201.427 194.177 186.927 179.675

5.191 4.782 4.410 4.068 3.754

2600 2700 2800 2900 3000

59.086 59.199 59.308 59.413 59.513

360.400 362.632 364.787 366.870 368.886

311.761 313.604 315.394 317.133 318.825

126.462 132.376 138.302 144.238 150.184

361.030 361.095 361.175 361.270 361.379

172.422 165.166 157.908 150.648 143.383

3.464 3.195 2.946 2.713 2.497

3100 3200 3300 3400 3500

59.609 59.706 59.794 59.881 59.969

370.839 372.733 374.572 376.358 378.095

320.471 322.075 323.638 325.162 326.650

156.140 162.106 168.081 174.065 180.057

361.502 361.638 361.786 361.946 362.118

136.114 128.842 121.565 114.283 106.996

2.294 2.103 1.924 1.756 1.597

3600 3700 3800 3900 4000

60.053 60.137 60.216 60.296 60.375

379.786 381.432 383.037 384.602 386.130

328.103 329.522 330.909 332.266 333.593

186.058 192.068 198.086 204.111 210.145

362.300 362.493 362.697 362.911 363.134

99.704 92.408 85.105 77.796 70.484

1.447 1.305 1.170 1.042 0.920

4100 4200 4300

60.450 60.530 60.605

387.621 389.079 390.504

334.893 336.166 337.413

216.186 222.235 228.292

363.368 363.611 363.864

63.164 55.840 48.508

0.805 0.694 0.589

T (K)

631

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Sulfur dioxide (SO2), ideal gas, molecular weight  64.0588 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

4400 4500

60.676 60.752

391.898 393.263

338.636 339.834

234.356 240.427

364.128 364.401

41.172 33.829

0.489 0.393

4600 4700 4800 4900 5000

60.823 60.894 60.969 61.036 61.107

394.599 395.908 397.190 398.448 399.682

341.010 342.185 343.298 341.410 345.504

246.506 252.592 258.685 264.785 270.893

364.686 364.981 365.288 365.607 365.938

26.480 19.125 11.763 4.394 2.981

0.301 0.213 0.128 0.047 0.031

5100 5200 5300 5400 5500

61.178 61.250 61.317 61.388 61.455

400.893 402.082 403.249 404.396 405.523

346.578 347.634 348.672 349.693 350.698

277.007 283.128 289.257 295.392 301.534

366.283 366.641 367.014 367.403 367.807

10.364 17.751 25.148 32.550 39.960

0.106 0.178 0.248 0.315 0.380

5600 5700 5800 5900 6000

61.522 61.588 61.655 61.727 61.793

406.631 407.720 408.792 409.846 410.884

351.687 352.661 353.619 354.563 355.493

307.683 313.838 320.000 326.169 332.346

368.230 368.671 369.131 369.611 370.113

47.379 54.804 62.238 69.677 77.128

0.442 0.502 0.561 0.617 0.671

log Kf

TABLE A2 Thermochemical Data of Selected Chemical Compounds Titanium dioxide (TiO2), crystal–liquid, molecular weight  79.8788 Enthalpy reference temperature  Tr  298.15 K [G°  H°(Tr)]/T (J/K/mol)

C°p (J/K/mol)

S° (J/K/mol)

0 100 200

0.000 18.502 42.012

0. 10.142 30.807

Infinite 89.638 54.969

298.15

55.103

50.292

500 600 700 800 900 1000

67.203 69.931 71.764 73.078 74.057 74.852

82.201 94.712 105.638 115.311 123.977 131.822

1100 1200 1300 1400 1500

75.479 76.023 76.525 76.944 77.320

138.986 145.577 151.683 157.370 162.691

T (K)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

8.636 7.950 4.833

939.870 942.649 944.360

939.870 Infinite 925.506 483.435 907.579 237.035

50.292

0.000

944.747

889.406 155.820

57.077 62.331 67.754 73.106 78.285 83.253

12.562 19.429 26.519 33.764 41.122 48.569

943.670 942.789 941.841 940.857 939.896 939.032

852.157 833.936 815.868 797.939 780.132 762.428

89.024 72.601 60.881 52.100 45.278 39.825

87.999 92.526 96.844 100.967 104.906

56.086 63.662 71.290 78.964 86.677

938.339 941.807 940.742 939.741 938.819

744.803 727.113 709.265 691.497 673.798

35.368 31.650 28.499 25.800 23.464

632

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Titanium dioxide (TiO2), crystal–liquid, molecular weight  79.8788 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

1600 1700 1800 1900 2000

77.655 77.990 78.283 78.576 78.868

167.692 172.410 176.876 181.117 185.155

108.676 112.287 115.753 119.082 122.286

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf 21.421 19.621 18.221 16.591 15.293

94.426 102.209 110.022 117.865 125.738

937.992 937.274 936.679 936.224 950.606

656.158 638.566 621.013 603.489 585.531

951.213

567.263 14.110

2100

79.161

189.010

125.372

133.640

2130 2130

79.228 100.416

190.133 221.563

126.276 126.276

136.016 202.960

2200 2300 2400 2500

100.416 100.416 100.416 100.416

224.809 229.273 233.547 237.646

129.360 133.608 137.683 141.601

209.989 220.030 230.072 240.114

883.396 881.909 880.444 879.001

551.188 536.122 521.120 506.177

13.087 12.176 11.342 10.576

2600 2700 2800 2900 3000

100.416 100.416 100.416 100.416 100.416

241.584 245.374 249.026 252.550 255.954

145.371 149.005 152.512 155.901 159.180

250.155 260.197 270.238 280.280 290.322

877.579 876.179 874.799 873.438 872.098

491.293 476.462 461.682 446.953 432.269

9.870 9.218 8.613 8.050 7.526

3100 3200 3300 3400 3500

100.416 100.416 100.416 100.416 100.416

259.247 262.435 265.525 268.522 271.433

162.355 165.433 168.420 171.320 174.139

300.363 310.405 320.446 330.488 340.530

870.775 869.471 868.184 866.914 865.660

417.630 403.034 388.478 373.960 359.480

7.037 6.579 6.149 5.745 5.365

3600 3700 3800 3900 4000

100.416 100.416 100.416 100.416 100.416

274.262 277.013 279.691 282.300 284.842

176.881 179.550 182.151 184.685 187.157

350.571 360.613 370.654 380.696 390.738

864.422 1272.296 1269.853 1267.478 1265.171

345.035 322.836 297.207 271.642 246.138

5.006 4.558 4.085 3.638 3.214

Crystal–liquid transition

TABLE A2 Thermochemical Data of Selected Chemical Compounds Titanium dioxide (TiO2), ideal gas, molecular weight  79.8788 Enthalpy reference temperature  Tr  298.15 K C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

0 100 200 250

0. 35.950 40.232 42.214

0. 217.025 243.358 252.547

298.15

44.149

260.148

T (K)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

Infinite 296.659 264.059 260.864

11.357 7.963 4.140 2.079

303.276 303.347 304.352 304.913

303.276 306.893 310.081 311.449

Infinite 160.304 80.985 65.074

260.148

0.

305.432

312.660

54.777

633

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Titanium dioxide (TiO2), ideal gas, molecular weight  79.8788 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

300 350 400 450 500

44.222 46.106 47.777 49.210 50.419

260.422 267.382 273.651 279.363 284.612

260.149 260.695 261.929 263.553 265.400

0.082 2.341 4.689 7.114 9.606

305.451 305.956 306.429 306.878 307.311

312.705 313.873 314.971 316.012 317.004

54.447 46.843 41.131 36.682 33.117

600 700 800 900 1000

52.280 53.591 54.532 55.222 55.740

293.979 302.143 309.363 315.828 321.674

269.402 273.509 277.548 281.448 285.183

14.746 20.044 25.452 30.942 36.491

308.156 309.001 309.854 310.761 311.795

318.863 320.581 322.177 323.664 325.043

27.759 23.922 21.036 16.785 16.979

1100 1200 1300 1400 1500

56.136 56.446 56.693 56.891 57.053

327.006 331.905 336.433 340.642 344.573

288.747 292.142 295.377 298.462 301.406

42.086 47.716 53.373 59.052 64.750

313.024 318.438 319.344 320.337 321.431

326.310 327.337 328.042 328.674 329.233

15.495 14.249 13.181 12.263 11.465

1600 1700 1800 1900 2000

57.188 57.300 57.396 57.478 57.550

348.259 351.730 355.008 356.113 361.063

304.220 306.914 309.496 311.973 314.355

70.462 76.187 81.922 87.666 93.417

322.641 323.980 325.465 327.108 343.612

329.714 330.116 330.435 330.667 330.354

10.764 10.143 9.589 9.091 8.628

2100 2200 2300 2400 2500

57.614 57.673 57.727 57.781 57.833

363.873 366.554 369.119 371.577 373.937

316.646 318.854 320.985 323.042 325.031

99.175 104.940 110.710 116.485 122.266

346.363 349.130 351.915 354.716 357.534

329.624 328.761 327.773 326.664 325.437

8.199 7.806 7.444 7.110 6.800

2600 2700 2800 2900 3000

57.888 57.945 58.006 58.072 58.144

376.206 378.392 380.500 382.537 384.507

326.956 328.820 330.629 332.384 334.088

128.052 133.843 139.641 145.445 151.256

360.368 363.217 366.081 368.959 371.849

324.098 322.648 321.093 319.436 317.679

6.511 6.242 5.990 5.754 5.531

3100 3200 3300 3400 3500

58.223 58.310 58.405 58.509 58.622

386.415 388.265 390.060 391.806 393.503

335.746 337.358 338.928 340.458 341.949

157.074 162.900 168.736 174.582 180.438

374.750 377.661 380.580 383.505 386.437

315.825 313.879 311.841 308.713 307.501

5.322 5.124 4.936 4.758 4.589

3600 3700 3800 3900 4000

58.743 58.874 59.013 59.161 59.318

395.156 396.768 398.339 399.874 401.374

343.404 344.825 346.213 347.569 348.895

186.306 192.187 198.081 203.990 209.914

389.371 801.407 803.110 804.869 806.679

305.204 295.037 281.328 267.574 253.775

4.428 4.165 3.867 3.584 3.314

4100 4200 4300 4400 4500

59.483 59.655 59.834 60.019 60.210

402.841 404.276 405.682 407.060 408.411

350.193 351.464 352.709 353.928 355.124

215.854 221.811 227.785 233.778 239.789

808.539 810.446 812.397 814.390 816.421

239.929 226.038 212.100 198.117 184.088

3.057 2.811 2.577 2.352 2.137

T (K)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

634

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Titanium dioxide (TiO2), ideal gas, molecular weight  79.8788 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

4600 4700 4800 4900 5000

60.406 60.607 60.812 61.019 61.229

409.736 411.037 412.315 413.571 414.806

356.297 357.448 358.578 359.687 360.777

245.820 251.871 257.941 264.033 270.145

818.489 820.591 822.725 824.889 827.081

170.014 155.894 141.729 127.518 113.264

1.931 1.733 1.542 1.359 1.183

5100 5200 5300 5400 5500

61.440 61.653 61.865 62.077 62.289

416.021 417.216 418.392 419.551 420.692

361.849 362.902 363.938 364.957 365.960

276.279 282.433 288.609 294.806 301.025

829.299 831.542 833.807 836.095 838.403

98.966 84.625 70.238 55.810 41.340

1.014 0.850 0.692 0.540 0.393

5600 5700 5800 5900 6000

62.498 62.706 62.911 63.113 63.311

421.816 422.924 424.016 425.093 426.156

366.947 367.920 368.877 369.821 370.751

307.264 313.524 319.805 326.106 332.428

840.731 843.079 845.444 847.828 850.230

26.826 12.272 2.325 16.961 31.640

0.250 0.112 0.021 0.150 0.275

log Kf

TABLE A2 Thermochemical Data of Selected Chemical Compounds Ozone (O3), ideal gas, molecular weight  47.9982 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

log Kf

0 100 200

0. 33.292 35.058

0. 200.791 224.221

Infinite 271.040 242.401

10.351 7.025 3.636

145.348 144.318 143.340

145.348 150.235 156.541

Infinite 78.474 40.884

298.15

39.238

238.932

238.932

0.000

142.674

163.184

28.589

300 400 500

39.330 43.744 47.262

239.175 251.116 261.272

238.933 240.531 243.688

0.073 4.234 8.792

142.665 142.370 142.340

163.311 170.247 177.224

28.435 22.232 18.514

600 700 800 900 1000

49.857 51.752 53.154 54.208 55.024

270.129 277.963 284.969 291.292 297.048

247.373 251.194 254.986 258.674 262.228

13.654 18.738 23.986 29.356 34.819

142.462 142.665 142.907 143.169 143.439

184.191 191.130 198.037 204.913 211.759

16.035 14.262 12.931 11.893 11.061

1100 1200 1300 1400 1500

55.660 56.174 56.593 56.948 57.245

302.323 307.189 311.702 315.909 319.849

265.637 268.899 272.020 275.007 277.866

40.355 45.947 51.586 57.264 62.974

143.711 143.980 144.245 144.502 144.750

218.578 225.372 232.144 238.896 245.629

10.379 9.810 9.328 8.913 8.554

T (K)

635

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Ozone (O3), ideal gas, molecular weight  47.9982 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

1600 1700 1800 1900 2000

57.501 57.722 57.919 58.095 58.250

323.551 327.044 330.349 333.485 336.469

280.607 283.237 285.763 288.193 290.533

68.711 74.473 80.255 86.056 91.873

144.987 145.211 145.419 145.611 145.784

252.347 259.050 265.740 272.419 279.089

8.238 7.960 7.712 7.489 7.289

2100 2200 2300 2400 2500

58.396 58.526 58.647 58.764 58.869

339.315 342.035 344.639 347.137 349.538

292.789 294.966 297.069 299.104 301.073

97.705 103.552 109.410 115.281 121.163

145.938 146.072 146.185 146.276 146.346

285.750 292.406 299.054 305.698 312.339

7.108 6.943 6.792 6.653 6.526

2600 2700 2800 2900 3000

58.969 59.066 59.158 59.245 59.329

351.849 354.077 356.226 358.304 360.314

302.982 304.833 306.631 308.377 310.075

127.055 132.956 138.868 144.788 150.716

146.393 146.419 146.424 146.408 146.371

318.978 325.616 332.253 338.889 345.527

6.408 6.299 6.198 6.104 6.016

3100 3200 3300 3400 3500

59.409 59.488 59.563 59.639 59.714

362.260 364.148 365.980 367.759 369.489

311.727 313.336 314.904 316.432 317.923

156.653 162.598 168.551 174.511 180.479

146.314 146.238 146.144 146.033 145.905

352.167 358.808 365.451 372.100 378.750

5.934 5.857 5.785 5.717 5.653

3600 3700 3800 3900 4000

59.781 59.852 59.919 59.986 60.053

371.172 372.811 374.408 375.965 377.485

319.379 320.801 322.191 323.550 324.879

186.453 192.435 198.424 204.419 210.421

145.761 145.601 145.427 145.239 145.038

385.405 392.063 398.728 405.396 412.069

5.592 5.535 5.481 5.430 5.381

4100 4200 4300 4400 4500

60.120 60.183 60.245 60.308 60.371

378.968 380.418 381.835 383.220 384.576

326.181 327.455 328.703 329.926 331.126

216.429 222.445 228.466 234.494 240.528

144.823 144.597 144.358 144.106 143.844

418.747 425.431 432.121 438.815 445.516

5.335 5.291 5.249 5.209 5.171

4600 4700 4800 4900 5000

60.434 60.492 60.555 60.614 60.672

385.904 387.204 388.479 389.728 390.953

332.302 333.457 334.590 335.702 336.795

246.568 252.614 258.667 264.725 270.789

143.568 143.281 142.981 142.667 142.340

452.222 458.935 465.655 472.381 479.113

5.135 5.100 5.067 5.036 5.005

5100 5200 5300 5400 5500

60.731 60.789 60.848 60.906 60.965

392.155 393.335 394.493 395.631 396.749

337.869 338.924 339.962 340.982 341.986

276.859 282.935 289.017 295.105 301.199

141.999 141.643 141.270 140.880 140.472

485.853 492.597 499.351 506.110 512.876

4.976 4.948 4.921 4.896 4.871

5600 5700 5800 5900 6000

61.024 61.078 61.137 61.191 61.250

397.848 398.929 399.992 401.037 402.066

342.974 343.946 344.903 345.846 346.774

307.298 313.403 319.514 325.630 331.752

140.045 139.596 139.126 138.631 138.111

519.653 526.434 533.226 540.023 546.832

4.847 4.824 4.802 4.781 4.761

log Kf

636

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Sulfur trioxide (SO3), ideal gas, molecular weight  80.0582 Enthalpy reference temperature  Tr  298.15 K C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

0 100 200 250

0. 34.076 42.336 46.784

0. 212.371 238.259 248.192

298.15

50.661

300 350 400 450 500

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

Infinite 295.976 261.145 257.582

11.697 8.361 4.577 2.348

390.025 391.735 393.960 394.937

390.025 385.724 378.839 374.943

Infinite 201.481 98.943 78.340

256.769

256.769

0.

395.765

371.016

65.000

50.802 54.423 57.672 60.559 63.100

257.083 265.191 272.674 279.637 286.152

256.770 257.402 258.849 260.777 262.992

0.094 2.726 5.530 8.487 11.580

395.794 396.543 399.412 400.656 401.878

370.862 366.646 362.242 357.529 352.668

64.573 54.719 47.304 41.501 36.843

600 700 800 900 1000

67.255 70.390 72.761 74.570 75.968

298.041 308.655 318.217 326.896 334.828

267.862 272.945 278.017 282.973 287.768

18.107 24.997 32.160 39.531 47.060

403.675 405.014 406.068 460.062 459.581

342.647 332.365 321.912 310.258 293.639

29.830 24.801 21.019 18.007 15.338

1100 1200 1300 1400 1500

77.065 77.937 78.639 79.212 79.685

342.122 348.866 355.133 360.983 366.465

292.382 296.811 301.060 305.133 309.041

54.714 62.466 70.296 78.189 86.135

459.063 458.521 457.968 457.413 456.863

277.069 260.548 244.073 227.640 211.247

13.157 11.341 9.807 8.493 7.356

1600 1700 1800 1900 2000

80.079 80.410 80.692 80.932 81.140

371.620 376.485 381.090 385.459 389.616

312.793 316.398 319.865 323.203 326.421

94.124 102.149 110.204 118.286 126.390

456.323 455.798 455.293 454.810 454.351

194.890 178.567 162.274 146.009 129.768

6.363 5.487 4.709 4.014 3.389

2100 2200 2300 2400 2500

81.319 81.476 81.614 81.735 81.843

393.579 397.366 400.990 404.466 407.805

329.525 332.523 335.422 338.227 340.944

134.513 142.653 150.807 158.975 167.154

453.919 453.514 453.137 452.790 452.472

113.549 97.350 81.170 65.006 48.855

2.824 2.311 1.843 1.415 1.021

2600 2700 2800 2900 3000

81.939 82.025 82.102 82.171 82.234

411.017 414.111 417.096 419.978 422.765

343.578 346.133 348.614 351.026 353.371

175.343 183.541 191.748 199.961 208.182

452.183 451.922 451.690 451.487 451.311

32.716 16.587 0.467 15.643 31.748

0.657 0.321 0.009 0.282 0.553

3100 3200 3300 3400 3500

82.290 82.342 82.389 82.432 82.472

425.462 428.076 430.610 433.070 435.460

355.653 357.876 360.042 362.153 364.214

216.408 224.640 232.876 241.117 249.363

451.161 451.038 450.940 450.866 450.817

47.849 63.943 80.034 96.124 112.210

0.806 1.044 1.267 1.477 1.675

3600 3700 3800 3900 4000

82.508 82.542 82.573 82.601 82.628

437.784 440.045 442.247 444.392 446.484

366.225 368.190 370.110 371.987 373.824

257.612 265.864 274.120 282.379 290.640

450.792 450.789 450.809 450.850 450.914

128.297 144.382 160.469 176.556 192.643

1.862 2.038 2.206 2.365 2.516

T (K)

637

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Sulfur trioxide (SO3), ideal gas, molecular weight  80.0582 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

4100 4200 4300 4400 4500

82.653 82.676 82.697 82.717 82.735

448.524 450.516 452.462 454.364 456.223

375.621 377.381 379.104 380.793 382.449

298.904 307.171 315.439 323.710 331.983

450.999 451.105 451.234 451.383 451.555

208.733 224.825 240.921 257.019 273.120

2.659 2.796 2.927 3.051 3.170

4600 4700 4800 4900 5000

82.753 82.769 82.785 82.799 82.813

458.041 459.821 461.564 463.271 464.944

384.072 385.665 387.228 388.763 390.270

340.257 348.533 356.811 365.090 373.371

451.749 451.965 452.205 452.469 452.757

289.225 305.336 321.452 337.573 353.700

3.284 3.393 3.498 3.599 3.695

5100 5200 5300 5400 5500

82.825 82.837 82.849 82.860 82.870

466.584 468.192 469.770 471.319 472.840

391.750 393.205 394.634 396.040 397.423

381.653 389.936 398.220 406.505 414.792

453.071 453.412 453.780 454.178 454.605

369.832 385.969 402.116 418.268 434.427

3.788 3.877 3.963 4.046 4.126

5600 5700 5800 5900 6000

82.879 82.889 82.897 82.906 82.913

474.333 475.800 477.242 478.659 480.052

398.783 400.121 401.439 402.735 404.012

423.079 431.368 439.657 447.947 456.238

455.065 455.557 456.084 456.647 457.248

450.597 466.773 482.960 499.153 515.359

4.203 4.277 4.350 4.419 4.487

log Kf

TABLE A2 Thermochemical Data of Selected Chemical Compounds Titanium oxide (Ti3O5), crystal–liquid, molecular weight  223.6370 Enthalpy reference temperature  Tr  298.15 K C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

0 100 200

0.000 46.116 114.445

0.000 21.062 75.659

298.15

154.808

300 400 450 450 500

155.477 182.841 189.954 181.586 184.096

600 700 800 900 1000

189.117 194.138 199.158 204.179 209.200

T (K)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

Infinite 237.166 142.352

23.108 21.610 13.339

2446.057 2453.501 2458.258

2446.057 Infinite 2409.803 1258.752 2363.911 617.390

129.369

129.369

0.000

2459.146

2317.293 405.980

130.329 179.228 201.217 230.691 249.952

129.372 135.852 141.909 141.909 151.765

0.287 17.350 26.688 39.952 49.094

2459.135 2316.413 403.323 2457.340 2269.052 296.308 2455.868 2245.602 260.663 Alpha–beta transition 2441.465 2223.774 232.316

283.961 313.490 339.742 363.489 385.261

171.037 189.323 206.514 222.657 237.844

67.754 86.917 106.582 126.749 147.418

2439.180 2436.818 2434.270 2431.589 2428.938

2180.450 2137.514 2094.928 2052.670 2010.711

log Kf

189.825 159.503 136.785 119.134 105.029

638

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Titanium oxide (Ti3O5), crystal–liquid, molecular weight  223.6370 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

1100 1200 1300 1400 1500

214.221 219.242 224.262 229.283 234.304

405.436 424.291 442.038 458.841 474.831

252.173 265.739 278.624 290.902 302.635

168.589 190.262 212.437 235.114 258.294

2426.485 2436.168 2431.892 2427.425 2422.799

1969.009 1927.162 1884.917 1843.008 1801.425

93.500 83.887 75.737 68.763 62.731

1600 1700 1800 1900 2000

239.325 243.346 249.366 254.387 259.408

490.113 504.773 518.881 532.496 545.674

313.878 324.679 335.079 345.113 354.813

281.975 306.159 330.844 356.032 381.722

2418.050 2413.214 2408.329 2403.433 2442.627

1760.154 1719.183 1678.499 1638.087 1596.570

57.463 52.824 48.709 45.034 41.698

2050 2050

261.918 267.776

552.111 635.791

359.547 359.547

394.755 566.299

2100 2200 2300 2400 2500

267.776 267.776 267.776 267.776 267.776

642.243 654.700 666.603 678.000 688.931

366.201 379.034 391.280 402.991 414.212

579.668 606.465 633.243 660.020 686.798

2268.296 2265.210 2262.180 2259.206 2256.287

1558.516 1524.788 1491.201 1457.745 1424.411

38.766 36.203 33.866 31.727 29.761

2600 2700 2800 2900 3000

267.776 267.776 267.776 267.776 267.776

699.433 709.539 719.278 728.674 737.752

424.981 435.334 445.302 454.913 464.190

713.576 740.353 767.131 793.908 820.686

2253.421 2250.608 2247.846 2245.133 2242.470

1391.193 1358.083 1325.077 1292.170 1259.354

27.949 26.274 24.720 23.275 21.927

3100 3200 3300 3400 3500

267.776 267.776 267.776 267.776 267.776

746.533 755.034 763.274 771.268 779.030

473.157 481.834 490.238 498.387 506.295

847.464 874.241 901.019 927.796 954.574

2239.852 2237.280 2234.751 2232.264 2229.818

1226.626 1193.984 1161.421 1128.932 1096.518

20.669 19.490 18.384 17.344 16.365

3600 3700 3800 3900 4000

267.776 267.776 267.776 267.776 267.776

786.574 793.910 801.051 808.007 814.787

513.976 521.443 528.708 535.780 542.671

981.352 1008.129 1034.907 1061.684 1088.462

2227.409 3452.333 3446.295 3440.457 3434.815

1064.171 1008.528 942.558 876.746 811.083

15.441 14.238 12.956 11.743 10.592

Beta–liquid transition

639

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Titanium oxide (Ti3O5), liquid, molecular weight  223.6370 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

0 100 200 298.15

173.962

232.446

232.446

0.000

2289.059

2177.939

381.566

300 400 500

174.054 179.075 184.096

233.522 284.282 324.781

232.449 239.336 252.507

0.322 17.978 36.137

2289.013 2286.625 2284.335

2177.250 2140.359 2190.058

379.093 279.502 219.809

600 700 800 900 1000

189.117 194.138 199.158 204.179 209.200

358.789 388.319 414.570 438.318 460.090

267.460 282.661 297.539 311.882 325.629

54.798 73.960 93.625 113.792 134.461

2282.050 2279.687 2277.139 2274.459 2271.808

2068.217 2032.764 1997.660 1962.886 1928.410

180.051 151.687 130.434 113.923 100.730

1100 1200 1300 1400

214.221 219.242 224.262 229.283

480.264 499.119 516.866 533.669

338.781 351.365 363.420 371.985

155.632 177.305 199.480 222.158

2269.355 2279.038 2274.762 2270.295

1894.191 1859.826 1825.064 1790.638

89.948 80.956 73.332 66.810

1400 1400

229.283 267.776

533.669 533.669

374.985 374.985

222.158 222.158

Glass–liquid transition

1500

267.776

552.144

386.187

248.935

2262.071

1756.666

61.173

1600 1700 1800 1900 2000

267.776 267.776 267.776 267.776 267.776

569.426 585.660 600.965 615.443 629.178

397.105 407.724 418.039 428.051 437.767

275.713 302.490 329.268 356.046 382.823

2254.226 2246.795 2239.818 2233.333 2271.438

1723.230 1690.273 1657.741 1625.582 1592.390

56.258 51.936 48.106 44.690 41.589

2050

267.776

635.790

442.516

396.212

2100 2200 2300 2400 2500

267.776 267.776 267.776 267.776 267.776

642.243 654.700 666.603 678.000 688.931

447.195 456.346 465.231 473.861 482.246

409.601 436.378 463.156 489.934 516.711

2268.296 2265.210 2262.180 2259.206 2256.287

1558.516 1524.788 1491.201 1457.745 1424.411

38.766 36.203 33.866 31.727 29.761

2600 2700 2800 2900 3000

267.776 267.776 267.776 267.776 267.776

699.433 709.539 719.278 728.674 737.752

490.399 498.329 506.048 513.563 520.886

543.489 570.266 597.044 623.822 650.599

2253.421 2250.608 2247.846 2245.133 2242.470

1391.193 1358.083 1325.077 1292.170 1259.354

27.949 26.274 24.720 23.275 21.927

3100 3200 3300 3400 3500

267.776 267.776 267.776 267.776 267.776

746.533 755.034 763.274 771.268 779.030

528.024 534.986 541.779 548.412 554.891

677.377 704.154 730.932 757.710 784.487

2239.852 2237.280 2234.751 2232.264 2229.818

1226.626 1193.984 1161.421 1128.932 1096.518

20.669 19.490 18.384 17.344 16.365

3600 3700 3800 3900 4000

267.776 267.776 267.776 267.776 267.776

786.574 793.910 801.051 808.007 814.787

561.222 567.412 573.467 579.392 585.193

811.265 838.042 864.820 891.598 918.375

2227.409 3452.333 3446.295 3440.457 3434.815

1064.171 1008.528 942.558 876.746 811.083

15.441 14.238 12.956 11.743 10.592

Beta–liquid transition

640

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Sulfur (S), alpha–beta–liquid, molecular weight  32.06 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) (kJ/mol)

ΔfH° (kJ/mol)

ΔfG° (kJ/mol)

0 100 200 250

0. 12.770 19.368 21.297

0. 12.522 23.637 28.179

Infinite 49.744 34.038 32.422

4.412 3.722 2.080 1.061

0. 0. 0. 0.

0. 0. 0. 0.

0. 0. 0. 0.

298.15

22.698

32.056

32.056

0.

0.

0.

0.

300 350

22.744 23.870

32.196 35.789

32.056 32.337

0.042 1.208

0. 0.

0. 0.

0. 0.

368.300 368.300

24.246 24.773

37.015 38.103

32.540 32.540

1.648 2.049

Alpha–beta transition

388.360 388.360

25.167 31.062

39.427 43.859

32.861 32.861

2.550 4.271

Beta–liquid transition

400

32.162

44.793

33.195

4.639

0.

432.020 432.020

53.808 53.806

47.431 47.431

34.151 34.151

5.737 5.737

Cp lambda maximum transition

450 500

43.046 37.986

49.308 53.532

34.720 36.398

6.564 8.567

0. 0.

0. 0.

0. 0.

600 700 800

34.308 32.681 31.699

60.078 65.241 69.530

39.825 43.099 46.143

12.152 15.499 18.710

0. 0. 0.

0. 0. 0.

0. 0. 0.

T (K)

log Kf

0.

0.

Fugacity  1 bar

882.117

31.665

72.624

48.467

21.310

900 1000

31.665 31.665

73.260 76.596

48.952 51.553

21.877 25.043

53.090 51.780

1.079 7.028

0.063 0.367

1100 1200 1300 1400 1500

31.665 31.665 31.665 31.665 31.665

79.614 82.369 84.904 87.250 89.435

53.969 56.222 58.332 60.315 62.185

28.209 31.376 34.542 37.709 40.875

50.485 49.205 47.941 46.693 45.460

12.846 18.546 24.141 29.639 35.048

0.610 0.807 0.970 1.106 1.220

641

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds

Sulfur, monatomic (S), ideal gas, molecular weight  32.06 Enthalpy reference temperature  Tr  298.15 K

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

0 100 200 250

0. 21.356 23.388 23.696

0. 142.891 158.392 163.653

Infinite 188.580 169.994 168.218

298.15

23.673

167.828

300 350 400 450 500

23.669 23.480 23.233 22.979 22.741

167.974 171.610 174.729 177.451 179.859

600 700 800 900 1000

22.338 22.031 21.800 21.624 21.489

1100 1200 1300 1400 1500

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

6.657 4.569 2.320 1.141

274.735 276.133 276.740 276.899

274.735 263.096 249.789 243.031

Infinite 137.427 65.238 50.779

167.828

0.

276.980

236.500

41.434

167.828 168.116 166.752 169.571 170.481

0.044 1.223 2.391 3.546 4.689

276.982 276.995 274.732 273.962 273.102

236.248 229.458 222.757 216.297 209.938

41.135 34.245 29.089 25.107 21.932

183.968 187.388 190.314 192.871 195.142

172.398 174.302 176.125 177.847 179.465

6.942 9.160 11.351 13.522 15.677

271.770 270.641 269.621 215.535 215.834

197.436 185.138 172.994 162.055 156.096

17.188 13.815 11.295 9.405 8.154

21.386 21.307 21.249 21.209 21.186

197.185 199.042 200.745 202.318 203.781

180.984 182.413 183.758 185.029 186.231

17.821 19.955 22.083 24.205 26.325

216.106 216.354 216.579 216.784 216.970

150.109 144.098 138.067 132.020 125.959

7.128 6.272 5.548 4.926 4.386

1600 1700 1800 1900 2000

21.178 21.184 21.203 21.234 21.276

205.148 206.432 207.643 208.790 209.880

187.371 188.455 189.487 190.473 191.417

28.443 30.561 32.680 34.802 36.927

217.140 217.295 217.438 217.570 217.694

119.886 113.803 107.711 101.611 95.505

3.914 3.497 3.126 2.793 2.494

2100 2200 2300 2400 2500

21.327 21.386 21.452 21.523 21.598

210.920 211.913 212.865 213.780 214.660

192.321 193.189 194.024 194.828 195.604

39.058 41.193 43.335 45.484 47.640

217.812 217.925 218.035 218.143 218.250

89.392 83.275 77.152 71.024 64.892

2.224 1.977 1.752 1.546 1.356

2600 2700 2800 2900 3000

21.676 21.756 21.837 21.919 21.999

215.508 216.328 217.121 217.888 218.633

196.353 197.078 197.780 198.460 199.120

49.803 51.975 54.155 56.343 58.538

218.358 218.467 218.579 218.694 218.811

58.755 52.615 46.470 40.321 34.169

1.180 1.018 0.867 0.726 0.595

3100 3200 3300 3400 3500

22.078 22.155 22.230 22.303 22.372

219.355 220.058 220.740 221.405 222.053

199.761 200.384 200.991 201.582 202.157

60.742 62.954 65.173 67.400 69.634

218.931 219.055 219.183 219.313 219.447

28.012 21.851 15.687 9.518 3.346

0.472 0.357 0.248 0.146 0.050

3600 3700 3800 3900 4000

22.439 22.502 22.561 22.618 22.670

222.684 223.300 223.900 224.487 225.061

202.719 203.267 203.802 204.325 204.836

71.874 74.121 76.375 78.634 80.898

219.583 219.721 219.862 220.003 220.146

2.830 9.011 15.194 21.382 27.573

0.041 0.127 0.209 0.286 0.360

642

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds

Sulfur, monatomic (S), ideal gas, molecular weight  32.06 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

4100 4200 4300 4400 4500

22.720 22.766 22.808 22.847 22.884

225.621 226.169 226.705 227.230 227.744

205.336 205.826 206.305 206.775 207.235

83.167 85.442 87.720 90.003 92.290

220.289 220.432 220.575 220.716 220.856

33.768 39.966 46.168 52.373 58.581

0.430 0.497 0.561 0.622 0.680

4600 4700 4800 4900 5000

22.917 22.947 22.974 22.999 23.021

228.247 228.740 229.224 229.698 230.163

207.686 208.129 208.563 208.990 209.409

94.580 96.873 99.169 101.468 103.769

220.993 221.127 221.258 221.386 221.509

64.792 71.006 77.223 83.443 89.665

0.736 0.789 0.840 0.890 0.937

5100 5200 5300 5400 5500

23.040 23.057 23.072 23.085 23.096

230.619 231.066 231.505 231.937 232.361

209.820 210.224 210.622 211.013 211.397

106.072 108.377 110.683 112.991 115.300

221.628 221.741 221.849 221.952 222.048

95.890 102.117 108.346 114.577 120.810

0.982 1.026 1.068 1.108 1.147

5600 5700 5800 5900 6000

23.105 23.112 23.117 23.121 23.124

232.777 233.186 233.588 233.983 234.372

211.775 212.147 212.513 212.874 213.229

117.610 119.921 122.232 124.544 126.857

222.138 222.222 222.299 222.368 222.431

127.044 133.281 139.518 145.757 151.997

1.185 1.221 1.256 1.290 1.323

T (K)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

TABLE A2 Thermochemical Data of Selected Chemical Compounds

Titanium (Ti), crystal–liquid, molecular weight  47.88 Enthalpy reference temperature  Tr  298.15 K C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

0. 100 200 250

0. 14.334 22.367 24.074

0. 8.261 21.227 26.414

Infinite 50.955 32.989 31.169

298.15

25.238

30.759

300 350 400 450 500

25.276 26.169 26.862 27.418 27.877

30.915 34.882 38.423 41.620 44.534

600 700 800 900 1000

28.596 29.135 29.472 30.454 32.074

49.683 54.134 58.039 61.561 64.848

T (K)

Standard state pressure  p°  0.1 MPa H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

4.830 4.269 2.352 1.189

0. 0. 0. 0.

0. 0. 0. 0.

0. 0. 0. 0.

30.759

0.

0.

0.

0.

30.760 31.071 31.772 32.692 33.733

0.047 1.334 2.660 4.018 5.401

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

35.973 38.257 40.490 42.639 44.697

8.226 11.114 14.039 17.030 20.151

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

643

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds

Titanium (Ti), crystal–liquid, molecular weight  47.88 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

1100

34.334

68.006

46.673

23.466

0.

1166 1166

36.175 29.245

70.058 73.636

47.938 47.936

25.791 29.963

Alpha–beta transition

1200 1300 1400 1500

29.459 30.175 31.023 32.003

74.479 76.864 79.131 81.304

48.679 50.756 52.702 54.537

30.961 33.941 37.000 40.150

0. 0. 0. 0.

0. 0. 0. 0.

0. 0. 0. 0.

1600 1700 1800 1900

33.115 34.359 35.736 37.244

83.404 85.448 87.451 89.422

56.276 57.932 59.517 61.039

43.405 46.778 50.281 53.929

0. 0. 0. 0.

0. 0. 0. 0.

0. 0. 0. 0.

1939 1939

37.868 47.237

90.186 97.481

61.617 61.617

55.394 69.540

Beta–liquid transition

2000

47.237

98.944

62.734

72.421

0.

0.

0.

2100 2200 2300 2400 2500

47.237 47.237 47.237 47.237 47.237

101.249 103.446 105.546 107.557 109.485

64.513 66.233 67.897 69.508 71.069

77.145 81.869 86.592 91.316 96.040

0. 0. 0. 0. 0.

0 0. 0. 0. 0.

0. 0. 0. 0. 0.

2600 2700 2800 2900 3000

47.237 47.237 47.237 47.237 47.237

111.338 113.120 114.838 116.496 118.097

72.582 74.051 75.477 76.863 78.211

100.764 105.487 110.211 114.935 119.659

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

3100 3200 3300 3400 3500

47.237 47.237 47.237 47.237 47.237

119.646 121.146 122.600 124.010 125.379

79.523 80.800 82.045 83.259 84.443

124.382 129.106 133.830 138.554 143.277

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

3600

47.237

126.710

85.598

148.001

0.

0.

0.

3630.956

47.237

127.114

85.951

0.149

3700 3800 3900 4000

47.237 47.237 47.237 47.237

128.004 129.264 130.491 131.687

86.727 87.830 88.908 89.963

152.725 157.449 162.172 166.896

409.098 407.864 406.685 405.560

7.788 19.039 30.258 41.447

0.110 0.262 0.405 0.541

4100 4200 4300 4400 4500

47.237 47.237 47.237 47.237 47.237

132.853 133.991 135.103 136.189 137.250

90.995 92.005 92.994 93.964 94.914

171.620 176.343 181.067 185.791 190.515

404.488 403.467 402.495 401.570 400.691

52.609 63.745 74.857 85.948 97.018

0.670 0.793 0.909 1.020 1.126

log Kf

0.

0.

Fugacity  1 bar

644

Combustion

TABLE A2 Thermochemical Data of Selected Chemical Compounds Titanium (Ti), ideal gas, molecular weight  47.88 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

0 100 200 250

0. 26.974 26.487 25.355

0. 151.246 170.126 175.913

Infinite 203.383 182.593 180.704

7.539 5.214 2.493 1.198

470.920 472.684 473.488 473.620

470.920 458.386 443.708 436.245

Infinite 239.436 115.885 91.148

298.15

24.430

180.297

180.297

0.

473.629

429.044

75.167

300 350 400 450 500

24.399 23.661 23.104 22.683 22.360

180.448 184.151 187.272 189.968 192.340

180.297 180.591 181.236 182.060 182.972

0.045 1.246 2.414 3.558 4.684

473.627 473.541 473.383 473.169 472.912

428.768 421.297 413.843 406.413 399.009

74.655 62.875 54.042 47.175 41.684

600 700 800 900 1000

21.913 21.632 21.454 21.353 21.323

196.374 199.729 202.605 205.126 207.373

184.881 186.769 188.573 190.275 191.875

6.896 9.072 11.226 13.366 15.499

472.299 471.587 470.816 469.965 468.977

384.284 369.670 355.163 340.756 326.451

33.455 27.585 23.190 19.777 17.052

1100 1200 1300 1400 1500

21.362 21.474 21.657 21.910 22.228

209.407 211.270 212.996 214.609 216.132

193.377 194.792 196.127 197.390 198.589

17.633 19.774 21.930 24.107 26.314

467.795 462.442 461.617 460.736 459.793

312.254 298.293 284.647 271.066 257.551

14.828 12.984 11.437 10.114 8.969

1600 1700 1800 1900 2000

22.604 23.029 23.497 23.999 24.529

217.578 218.961 220.290 221.574 222.818

199.731 200.822 201.867 202.870 203.837

28.555 30.836 33.162 35.537 37.963

458.779 457.688 456.510 455.237 439.171

244.101 230.717 217.399 204.149 191.423

7.969 7.089 6.309 5.612 4.999

2100 2200 2300 2400 2500

25.080 25.648 26.228 26.819 27.416

224.028 225.208 226.361 227.489 228.596

204.769 205.672 206.546 207.396 208.221

40.443 42.979 45.573 48.225 50.937

436.927 434.740 432.610 430.538 428.526

179.091 166.864 154.736 142.699 130.748

4.455 3.962 3.514 3.106 2.732

2600 2700 2800 2900 3000

28.019 28.626 29.235 29.846 30.456

229.683 230.752 231.804 232.841 233.863

209.026 209.811 210.578 211.328 212.062

53.709 56.541 59.434 62.388 65.403

426.574 424.683 422.852 421.082 419.374

118.875 107.077 95.347 83.682 72.077

2.388 2.072 1.779 1.507 1.255

3100 3200 3300 3400 3500

31.065 31.671 32.273 32.870 33.460

234.871 235.867 236.851 237.823 238.785

212.781 213.487 214.180 214.861 215.531

68.479 71.616 74.813 78.071 81.367

417.726 416.139 414.612 413.146 411.739

60.528 49.031 37.582 26.179 14.818

1.020 0.800 0.595 0.402 0.221

3600

34.042

239.736

216.190

84.762

410.390

3.497

0.051

3630.956

34.219

240.028

216.392

85.819

3700 3800

34.613 35.173

240.676 241.607

216.840 217.479

88.195 91.685

T (K)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

log Kf

Fugacity  1 bar 0. 0.

0. 0.

0. 0.

645

Appendix A: Thermochemical Data and Conversion Factors

TABLE A2 Thermochemical Data of Selected Chemical Compounds Titanium (Ti), ideal gas, molecular weight  47.88 Enthalpy reference temperature  Tr  298.15 K

Standard state pressure  p°  0.1 MPa

T (K)

C°p (J/K/mol)

S° (J/K/mol)

[G°  H°(Tr)]/T (J/K/mol)

H°  H°(Tr) ΔfH° (kJ/mol) (kJ/mol)

ΔfG° (kJ/mol)

3900 4000

35.721 36.254

242.527 243.439

218.110 218.731

95.229 98.828

0. 0.

0. 0.

0. 0.

4100 4200 4300 4400 4500

36.772 37.273 37.757 38.223 38.670

244.340 245.232 246.115 246.988 247.852

219.345 219.951 220.549 221.140 221.724

102.480 106.182 109.934 113.733 117.578

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

4600 4700 4800 4900 5000

39.097 39.505 39.892 40.259 40.606

248.707 249.552 250.388 251.214 252.031

222.301 222.872 223.437 223.995 224.548

121.466 125.397 129.367 133.374 137.418

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

5100 5200 5300 5400 5500

40.933 41.240 41.528 41.796 42.046

252.839 253.637 254.425 255.204 255.973

225.095 225.636 226.172 226.702 227.227

141.495 145.604 149.742 153.909 158.101

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

5600 5700 5800 5900 6000

42.277 42.490 42.686 42.866 43.029

256.733 257.483 258.224 258.955 259.677

227.747 228.263 228.773 229.278 229.779

162.317 166.556 170.815 175.092 179.387

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

log Kf

646

TABLE A3 Thermochemical Data for Species Included in Reaction List of Appendix C Species

ΔHf 298 (kJ/mol)

S298 (J/K/mol)

Cp 300 (J/K/mol)

Cp 400 (J/K/mol)

Cp 500 (J/K/mol)

Cp 600 (J/K/mol)

Cp 800 (J/K/mol)

Cp 1000 (J/K/mol)

Cp 1500 (J/K/mol)

0.00

154.85

20.79

20.79

20.79

20.79

20.79

20.79

20.79

55.69

59.83

62.84

68.33

Ar

Argon

H

Hydrogen

Table A2

O

Oxygen

Table A2

OH

Hydroxyl

H2

Hydrogen

Table A2

O2

Oxygen

Table A2

H2O

Water

Table A2

HO2

Hydroperoxyl

12.55

H2O2

Hydrogen peroxide

136.11

CO

Carbon monoxide

Table A2

CO2

Carbon dioxide

Table A2

HCO

Formyl

43.51

224.65

34.63

36.53

38.73

40.97

44.96

48.06

52.65

CH2O

Formaldehyde

115.90

218.95

35.46

39.26

43.77

48.22

55.98

62.00

71.18

HCOOH

Formic acid

387.50

248.01

44.82

53.52

60.85

67.03

76.76

83.96

94.80

CH2OH

Hydroxy methyl

17.78

246.35

47.38

54.14

60.16

65.37

73.37

78.62

87.65

37.24

Table A2

Table A2 232.99

43.22

48.45

52.55

Combustion

Methoxy

16.30

228.48

37.99

45.13

52.02

58.47

69.58

77.81

90.00

CH3OH

Methanol

201.10

239.65

43.98

51.90

59.64

66.98

79.80

89.53

104.66

C

Carbon

CH

Methylidyne

594.17

182.93

29.09

29.30

29.49

29.77

30.82

32.56

36.61

CH2

Methylene

386.99

195.48

34.50

35.78

37.15

38.60

41.57

44.21

49.11

1

Singlet methylene

424.72

188.71

33.79

34.72

36.00

37.58

41.20

44.38

49.48

CH3

Methyl

145.69

194.17

38.75

42.04

45.25

48.29

53.93

58.95

68.16

CH4

Methane

C2H

Ethynyl

44.87

48.27

50.94

55.69

C2H2

Acetylene

C2H3

Vinyl

59.88

71.04

78.43

88.97

C2H4

Ethene

C2H5

Ethyl

117.23

242.64

47.37

56.90

66.73

76.54

94.48

106.71

123.66

C2H6

Ethane

83.86

228.98

52.61

65.66

77.89

89.10

108.05

122.59

144.83

HCCO

Ketyl

177.60

254.16

52.93

56.34

59.53

62.44

67.25

70.43

75.24

HCCOH

Ethynol

85.49

245.64

55.30

61.84

67.62

72.59

80.10

84.94

93.24

CH2CO

Ketene

51.87

241.81

52.01

59.30

65.54

70.76

78.63

84.70

93.89

CH3CO

Acetyl

22.60

266.71

51.95

60.40

68.32

75.55

87.60

96.51

109.58

CH2

Table A2

Table A2 564.90

207.37

37.25

40.29

42.78 Table A2

286.25

231.52

40.03

46.81

53.48 Table A2

647

(continues)

Appendix A: Thermochemical Data and Conversion Factors

CH3O

648

TABLE A3 (continued) Species

ΔHf 298 (kJ/mol)

S298 (J/K/mol)

Cp 300 (J/K/mol)

Cp 400 (J/K/mol)

Cp 500 (J/K/mol)

Cp 600 (J/K/mol)

Cp 800 (J/K/mol)

Cp 1000 (J/K/mol)

Cp 1500 (J/K/mol)

Formyl methylene

14.71

258.90

54.31

65.84

75.29

83.06

94.91

103.53

117.06

CH3CHO

Acetaldehyde

165.32

263.82

55.43

66.41

76.61

85.84

101.11

112.48

129.17

C2H4O

Oxirane

52.64

243.01

48.16

62.40

75.41

86.31

102.97

114.96

133.16

C3H2

Propynylidene

542.29

271.22

62.47

67.36

70.74

73.41

78.33

82.59

88.76

C3H3

Propargyl

347.47

257.28

66.27

74.24

81.48

87.89

98.02

104.62

115.26

C3H4

Allene

199.31

242.45

59.61

70.98

81.41

90.83

106.47

117.98

134.12

pC3H4

Propyne

191.51

246.45

60.73

71.36

81.19

90.14

105.27

116.72

133.03

cC3H4

Cyclo-propene

284.53

242.50

53.39

66.39

78.19

88.71

105.77

117.76

133.50

C3H5

Allyl

161.70

270.91

67.24

81.78

95.04

106.82

125.50

137.60

156.62

CH3CCH2

iso-propenyl

255.62

289.77

64.66

76.89

88.75

99.96

119.25

132.23

151.02

CHCHCH3

Propenyl

270.94

287.65

65.03

77.65

89.70

100.92

119.89

132.54

151.18

C3H6

Propene

20.46

257.40

64.70

80.61

95.10

107.96

128.80

144.44

167.94

iC3H7

iso-propyl

76.15

251.50

75.35

93.09

108.88

122.61

144.30

160.74

185.80

nC3H7

n-propyl

94.56

268.37

75.74

93.16

108.72

122.28

143.81

160.22

185.38

C3H8

Propane

103.85

270.18

73.94

94.09

112.40

128.63

154.85

174.60

204.32

Combustion

CH2CHO

Nitrogen

Table A2

N2

Nitrogen

Table A2

NO

Nitric oxide

Table A2

NO2

Nitrogen dioxide

Table A2

N2O

Nitrous oxide

82.05

219.96

38.70

42.68

45.83

48.39

52.24

54.87

58.41

HNO

Nitrosyl hydride

106.26

220.72

34.67

36.77

39.09

41.29

45.04

47.92

52.35

HONO

Nitrous acid

76.74

249.42

45.47

51.36

56.12

59.90

65.42

69.24

74.96

NH

Imidogen

356.91

181.22

26.16

27.04

27.87

28.64

30.04

31.27

33.75

NH2

Amidogen

129.05

194.99

30.90

33.06

35.10

37.01

40.49

43.54

49.61

NH3

Ammonia

NNH

Diazenyl

249.51

224.50

37.78

39.63

41.34

42.91

45.69

48.01

52.19

N2H2

Diazene

212.97

218.60

36.62

40.94

45.51

49.71

56.68

62.05

70.66

N2H3

Hydrazyl

153.92

228.54

44.33

51.21

57.68

63.46

72.56

79.22

90.24

N2H4

Hydrazine

95.35

238.72

51.02

61.70

70.54

77.56

88.20

96.36

110.32

NO3

Nitrogen trioxide

71.13

252.62

47.11

55.93

62.60

67.38

73.27

76.48

80.01

HNO3

Nitric acid

134.31

266.40

53.53

63.19

70.84

76.77

85.04

90.43

97.98

Cl

Chlorine

121.30

165.19

21.85

22.47

22.74

22.78

22.55

22.23

21.65

Table A2

Appendix A: Thermochemical Data and Conversion Factors

N

(continues)

649

650

TABLE A3 (continued) Species

ΔHf 298 (kJ/mol)

S298 (J/K/mol)

Cp 300 (J/K/mol)

Cp 400 (J/K/mol)

Cp 500 (J/K/mol)

Cp 600 (J/K/mol)

Cp 800 (J/K/mol)

Cp 1000 (J/K/mol)

Cp 1500 (J/K/mol)

Cl2

Chlorine

0.00

223.08

33.98

35.30

36.06

37.55

37.11

37.44

37.95

HCl

Hydrogen chloride

92.31

186.90

29.14

29.18

29.30

29.58

30.49

31.63

34.06

ClO

Chlorine oxide

101.22

226.65

31.58

33.23

34.43

35.27

36.30

36.88

37.65

HOCl

Hydrochlorous acid

74.48

236.50

37.34

40.08

42.31

44.05

46.60

48.54

51.98

COCl

Carbonyl chloride

16.74

265.97

45.12

47.27

48.91

50.30

52.48

54.00

56.02

O3

Ozone

Table A2

S

Sulfur

Table A2

SO

Sulfur oxide

Table A2

SO2

Sulfur dioxide

Table A2

SO3

Sulfur trioxide

Table A2

SH

Mercapto

139.33

195.63

32.43

31.71

31.28

31.23

31.84

32.81

34.92

H2S

Hydrogen sulfide

20.50

205.76

34.21

35.58

37.19

38.94

42.52

45.79

51.48

HSO

Mercapto-oxy

22.59

241.84

37.74

41.55

44.89

47.53

51.13

53.26

55.81

Combustion

Hydroxy sulfur

0.00

239.12

36.44

39.41

41.84

43.72

46.36

48.24

51.59

HSO2

Hydrogen sulfonyl

141.42

266.68

49.96

57.24

62.72

66.82

72.30

75.52

79.41

HOSO

Hydrogen sulfinyl

241.42

270.62

49.66

56.19

60.92

64.27

68.62

71.38

75.69

HSOH

Oxadisulfane

119.24

245.43

45.31

51.13

56.36

60.50

66.19

69.54

73.64

H2SO

Hydrogen sulfoxide

47.28

239.58

39.87

46.57

52.97

58.37

66.19

71.13

77.15

HOSHO

Hydrogen sulfinic acid

269.87

269.78

56.82

66.27

73.85

79.54

87.11

91.84

98.49

HOSO2

Hydroxy sulfonyl

391.20

295.89

70.04

78.49

84.22

88.16

92.97

95.98

100.50

S2

Disulfur

128.4

228.00

32.59

34.01

35.06

35.81

36.65

37.20

38.87

HS2

Thiosulfeno

104.6

253.30

39.75

42.72

45.02

46.82

49.50

51.55

54.39

H2S2

Disulfane

15.9

251.00

48.87

54.18

58.16

61.25

66.07

69.79

75.40

Appendix A: Thermochemical Data and Conversion Factors

HOS

651

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Appendix B

Adiabatic Flame Temperatures of Hydrocarbons The adiabatic flame temperatures presented in Table B1 were calculated using the CEA code of Gordon and McBride (http://www.grc.nasa.gov/WWW/CEAWeb/) described in Appendix I for a constant pressure and constant enthalpy condition (initial temperature of 298 K and pressure of 1 atm). The mixture for each fuel is stoichiometric with air. The heats of formation for the hydrocarbons are from the CRC Handbook of Chemistry and Physics (82nd edition, CRC Press, Inc., Boca Raton, FL, 2002). TABLE B1 Adiabatic Flame Temperatures Chemical Formula

Name

CH2O

ΔHf,298 (kJ/mol)

Tad (K)

Formaldehyde

108.6

2374

CH3NO2

Nitromethane

112.6

2544

CH3NO3

Methyl nitrate

156.3

2732

CH3OH

Methanol

239.2

2150

CH4

Methane

74.6

2226

C2H2

Acetylene

C2H2O

Ketene

227.4

2541

67.9

2390

C2H4

Ethylene

52.4

2370

C2H4O C2H4O

Acetaldehyde

192.2

2258

Ethylene oxide

78

2376

C2H5NO2 C2H5NO3

Nitroethane

143.9

2407

Ethyl nitrate

190.4

2494

C2H6

Ethane

84.0

2260

C2H6O

Ethanol

277.6

2195

C2H6O

Dimethyl ether

203.3

2267

C3H4

Allene

190.5

2442 (continues)

653

654

Combustion

TABLE B1 (continued) Chemical Formula

Name

C3H4

Propyne

184.9

2439

C3H4

Cyclopropene

277.1

2494

C3H6

Propene

4.0

2325

C3H6

Cyclopropane

35.2

2344

C3H6O

Allyl achohol

171.8

2295

C3H6O

Propanal

215.6

2265

C3H6O

Acetone

248.4

2241

C3H7NO2

Nitropropane

167.2

2364

C3H7NO3

Propyl nitrate

214.5

2421

C3H8

Propane

120.9

2257

C3H8O

1-Propanol

302.6

2221

C4H6

1,2-Butadiene

138.6

2386

C4H6

1,3-Butadiene

88.5

2362

C4H6

1-Butyne

141.4

2388

C4H6

2-Butyne

119.1

2377

C4H6

Cyclobutene

156.7

2395

C4H8

1-Butene

20.8

2312

C4H8

cis-2-Butene

29.8

2308

C4H8

Isobutene

37.5

2304

C4H8

Cyclobutane

3.7

2324

C4H8

Methylcyclopropane

1.7

2323

C4H8O

Ethyl vinyl ether

167.4

2304

C4H8O

Butanal

239.2

2268

C4H9NO2

Nitrobutane

192.5

2342

C4H10

Butane

147.3

2260

C4H10

Isobutane

154.2

2257

C4H10O

1-Butanol

327.2

2234

C4H10O

2-Methyl-1-propanol

334.7

2230

ΔHf,298 (kJ/mol)

Tad (K)

655

Appendix B: Adiabatic Flame Temperatures of Hydrocarbons

TABLE B1 (continued) Chemical Formula

Name

C4H10O

Diethyl ether

C5H6

ΔHf,298 (kJ/mol)

Tad (K)

279.5

2257

1,3-Cyclopentadiene

105.9

2360

C5H8

1,2-Pentadiene

140.7

2374

C5H8

Cyclopentene

4.3

2320

C5H10

1-Pentene

46.9

2304

C5H10

cis-2-Pentene

53.7

2302

C5H10

Cyclopentene

105.1

2281

C5H10

Methylcyclobutane

44.5

2305

C5H12

Pentane

173.5

2262

C5H12

Isopentane

178.4

2260

C5H12

Neopentane

190.2

2256

C5H12O

1-Pentanol

351.6

2242

C5H12O

Tert-butyl methyl ether

313.6

2257

C6H5NO2

Nitrobenzene

12.5

2406

C6H6

1,5-Hexadiyene

384.2

2448

C6H6

Benzene

49.1

2331

C6H10

Cyclohexane

38.5

2306

C6H10

1-Methylcyclopentane

36.4

2307

C6H12

1-Hexene

74.2

2298

C6H12

cis-2-Hexene

83.9

2295

C6H12

2-Methyl-1-penetene

90.0

2293

C6H12

Cyclohexane

156.4

2271

C6H12

Methylcyclopentane

137.9

2278

C6H12

Ethylcyclobutane

59.0

2303

C6H14

Hexane

198.7

2264

C6H14

2-Methylpentane

204.6

2262

C6H14

2,2-Dimethylbutane

213.8

2259

C7H8

Toluene

12.4

2317 (continues)

656

Combustion

TABLE B1 (continued) Chemical Formula

Name

C7H14

1-Heptene

C7H14

ΔHf,298 (kJ/mol)

Tad (K)

97.9

2295

cis-2-Heptene

105.1

2293

C7H14

Cyclcoheptane

156.6

2279

C7H14

Methylcyclcohexane

190.1

2269

C7H14

Ethylcyclopentane

163.4

2277

C7H16

Heptane

224.2

2265

C7H16

2-Methylhexane

229.5

2264

C8H8

Styrene

103.8

2281

C8H10

1,7-Octadiyne

334.4

2402

C8H10

Ethylbenzene

12.3

2311

C8H10

o-Xylene

24.4

2308

C8H10

m-Xylene

25.4

2308

C8H10

p-Xylene

24.4

2308

C8H16

1-Octene

121.8

2293

C8H16

Cyclooctane

167.7

2282

C8H16

Ethylcyclohexane

212.1

2271

C8H16

Propylcyclopentane

188.8

2276

C8H18

Octane

250.1

2266

C8H18

2-Methylheptane

255.0

2265

C8H18

3-Ethylhexane

250.4

2266

C8H18

2,2-Dimethylhexane

261.9

2263

C9H8

Indene

110.6

2339

C9H10

Indan

11.5

2316

C9H12

Propylbenzene

38.3

2306

C9H12

Cumene

41.1

2305

C9H12

o-Ethyltoluene

46.4

2304

C9H18

Propylcyclohexane

237.4

2271

C9H20

Nonane

274.7

2267

657

Appendix B: Adiabatic Flame Temperatures of Hydrocarbons

TABLE B1 (continued) Chemical Formula

Name

C10H14

Butylbenzene

63.2

2303

C10H16

Dipentene

50.8

2308

C10H20

1-Decene

173.8

2289

C10H22

Decane

300.9

2267

C11H10

1-Methylnaphthalene

56.3

2323

C11H24

Undecane

327.2

2268

C12H26

Dodecane

350.9

2268

C16H26

Decylbenzene

218.3

2290

C16H32

1-Hexadecene

328.7

2282

C16H34

Hexadecane

456.1

2269

ΔHf,298 (kJ/mol)

Tad (K)

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Appendix C

Specific Reaction Rate Constants The rate constant data for the reactions that follow in this appendix are presented as sets of chemical mechanisms for describing high temperature oxidation of various fuels. The order of their presentation follows the hierarchical approach to combustion modeling described by Westbrook and Dryer [Prog. Energy Combust. Sci., 10, 1(1984)]. In this approach, reaction mechanisms are developed systematically, beginning with the simplest species (fuels) and reactions that are common sub-elements in the combustion of more complex species, and sequentially constructed by incorporating new species and reactions in order of increasing complexity. For example, the H2/O2 mechanism is a submechanism of the CO/H2/O2 mechanism, which is a submechanism of the CH2/CO/H2/O2 mechanism, which are all submechanisms of a methanol or methane oxidation mechanism. Tables C1-C6 provide examples of oxidation mechanisms for H2, CO, CH2O, CH3OH, CH4, and C2H6, respectively. References for the origin of each mechanism are listed at the end of each table. The reader should refer to these references for the original source of each rate constant quoted. The backward rate constant at a given temperature is determined through the equilibrium constant at the temperature. The units are in cm3 mol s kJ for the expression k  AT n exp(E/RT).

TABLE C1 H2/O2 Mechanisma A

n

E

0.40

69.45

H2ˆO2 Chain Reactions 1.1

H  O2  O  OH

3.55  1015

1.2

O  H2  H  OH

5.08  10

2.70

26.32

1.3

OH  H2  H  H2O

2.16  10

1.50

14.35

1.4

O  H2O  OH  OH

2.97  10

2.00

56.07

4 8 6

(continues)

659

660

Combustion

TABLE C1 (continued) A

n

E

4.58  1019

1.40

436.75

H2  Ar  H  H  Ar

5.84  1018

1.10

436.75

O  OM  O2  M

6.16  10

15

0.50

0.00

O  OAr  O2  Ar

1.89  1013

0.00

7.49

O  H  M  OH  M

4.71  1018

1.00

0.00

3.80  1022

2.00

0.00

H  O2  HO2, k

1.48  1012

0.60

0.00

H  O2  M  HO2  M, M  N2, k0

6.37  1020

1.72

2.18

9.04  1019

1.50

2.05

H2ˆO2 Dissociation/Recombination Reactions 1.5b

H2  M  H  H  M εH2  2.5, εH2O  12.0, εCO  1.9, εCO2  3.8, εAr  0.0

b

1.6

εH2  2.5, εH2O  12.0, εCO  1.9, εCO2  3.8, εAr  0.0

1.7b

εH2  2.5, εH2O  12.0, εCO  1.9, εCO2  3.8, εAr  0.75 1.8b

H  OH  M  H2O  M εH2  2.5, εH2O  12.0, εCO  1.9, εCO2  3.8, εAr  0.38

HO2 Reactions 1.9c

α  0.8, T***  1.0  1030, T*  1.0  1030 εH2  2, εH2O  11, εCO  1.9, εCO2  3.8, εO2  0.78 H  O2  M  HO2  M, M  Ar, k0 30

α  0.5, T  1.0  10 T*  1.0  1030 ***

,

εH2  3, εH2O  16, εCO  2.7, εCO2  5.4, εO2  1.1 1.10

HO2  H  H2  O2

1.66  1013

0.00

3.43

1.11

HO2  H  OH  OH

7.08  10

0.00

1.26

13

661

Appendix C: Specific Reaction Rate Constants

TABLE C1 (continued) A

n

E

1.12

HO2  O  OH  O2

3.25  1013

0.00

0.00

1.13

HO2  OH  H2O  O2

2.89  10

0.00

2.09

13

H2O2 Reactions 1.14d

c

1.15

HO2  HO2  H2O2  O2

4.20  1014

0.00

50.12

1.30  10

11

0.00

6.82

H2O2  OH  OH, k

2.95  10

14

0.00

202.51

H2O2  M  OH  OH  M, k0

1.20  10

17

0.00

190.37

30

α  0.5, T  1.0  10 T*  1.0  1030 ***

,

εH2  2.5, εH2O  12, εCO  1.9, εCO2  3.8, εAr  0.64 1.16

H2O2  H  H2O  OH

2.41  1013

0.00

16.61

1.17

H2O2  H  H2  HO2

4.82  1013

0.00

33.26

1.18

H2O2  O  OH  HO2

9.55  106

2.00

16.61

1.19d

H2O2  OH  H2O  HO2

1.00  1012

0.00

0.00

5.80  10

0.00

40.00

14

Sources: Li, J., Zhao, Z., Kazakov, A., Chaos, M., Dryer, F. L. , and Scire, J. J., Jr., A Comprehensive Kinetic Mechanism for CO, CH2O, and CH3OH Combustion, Int. J. Chem. Kinet. 39, 109–136 (2007); Li, J., Zhao, Z., Kazakov, A., and Dryer, F. L., An Updated Comprehensive Kinetic Model of Hydrogen Combustion, Int. J. Chem. Kinet. 36, 565 (2004). Reaction rates in cm3 mol s kJ units, k  ATn exp(E/RT). ki  εi  kM, εi  1 for chemical species not defined. c The fall-off behavior of this reaction is expressed as k  k[Pr/(1  Pr)]F, Pr  k0[M]/ k, log(F)  [1  [(log Pr  c)/(n  d{log Pr  c})]2]1 log Fcent, c  0.4  0.67 log Fcent, n  0.75  1.27 log Fcent, d  0.14, and Fcent  (1  α)exp(T/T***  α exp(T/T*), Gilbert, R. G., Luther, K., and Troe, J., Ber. Bunsenges. Phys. Chem. 87, 169(1983). d Rate represented by sum of two Arrhenius expressions, using the Arrhenius parameters of the current reaction line and the line below. a b

TABLE C2 CO/H2/O2 Mechanisma A

n

E

1.80  1010

0.00

9.98

CO and CO2 Reactions 2.1b

CO  O  CO2, k

(continues)

662

Combustion

TABLE C2 (continued) A CO  O  M  CO2  M, k0

n

1.55  10

E

24

2.79

17.54

εH2  2.5, εH2O  12, εCO  1.9, εCO2  3.8, εAr  0.87 2.2

CO  O2  CO2  O

2.53  1012

0.00

199.53

2.3

CO  OH  CO2  H

2.23  105

1.90

4.85

2.4

CO  HO2  CO2  OH

3.01  1013

0.00

96.23

4.75  1011

0.706

62.34

HCO Reactions 2.5b

HCO  M  H  CO  M εH2  2.5, εH2O  6, εCO  1.9, εCO2  3.8

2.6

HCO  O2  CO  HO2

7.58  1012

0.00

1.72

2.7

HCO  H  CO  H2

7.23  10

13

0.00

0.00

2.8

HCO  O  CO  OH

3.02  10

13

0.00

0.00

2.9

HCO  O  CO2  H

3.00  10

0.00

0.00

2.10

HCO  OH  CO  H2O

3.02  10

0.00

0.00

2.11

HCO  HO2  CO2  OH  H

3.00  10

0.00

0.00

2.12

HCO  HCO  H2  CO  CO

3.00  10

0.00

0.00

13 13 13 12

Source: Li, J., Zhao, Z., Kazakov, A., Chaos, M., Dryer, F. L., and Scire, J. J., Jr., A Comprehensive Kinetic Mechanism for CO, CH2O, and CH3OH Combustion, Int. J. Chem. Kinet. 39, 109–136 (2007). a Reaction rates in cm3 mol s kJ units, k  ATn exp(E/RT). b ki  εi  kM, εi  1 for chemical species not defined.

TABLE C3 CH2O/CO/H2/O2 Mechanisma A

n

E

6.30

418.00

CH2O Reactions 3.1b

CH2O  M  HCO  H  M εH2  2.5, εH2O  12, εCO  1.9, εCO2  3.8, εAr  0.87

3.30  1039

663

Appendix C: Specific Reaction Rate Constants

TABLE C3 (continued) A b

3.2

CH2O  M  CO  H2  M

3.10  10

45

n

E

8.00

408.00

εH2  2.5, εH2O  12, εCO  1.9, εCO2  3.8, εAr  0.87 3.3

CH2O  H  HCO  H2

5.74  107

1.90

11.50

3.4

CH2O  O  HCO  OH

1.81  1013

0.00

12.89

3.5

CH2O  OH  HCO  H2O

3.43  109

1.20

1.88

3.6

CH2O  O2  HCO  HO2

1.23  106

3.00

217.60

3.7

CH2O  HO2  HCO  H2O2

4.11  10

4

2.50

42.72

3.8

HCO  HCO  CH2O  CO

3.00  10

13

0.00

0.00

Source: Li, J., Zhao, Z., Kazakov, A., Chaos, M., Dryer, F. L., and Scire, J. J., Jr., A Comprehensive Kinetic Mechanism for CO, CH2O, and CH3OH Combustion, Int. J. Chem. Kinet. 39, 109–136 (2007). a Reaction rates in cm3 mol s kJ units, k  ATn exp(E/RT). b ki  εi  kM, εi  1 for chemical species not defined.

TABLE C4 CH3OH/CH2O/CO/H2/O2 Mechanisma,e A

n

E

CH2OH Reactions 4.1

CH2OH  M  CH2O  H  M

1.00  1014

0.00

105.00

4.2

CH2OH  H  CH2O  H2

6.00  10

12

0.00

0.00

4.3

CH2OH  O  CH2O  OH

4.20  10

13

0.00

0.00

4.4

CH2OH  OH  CH2O  H2O

2.40  10

13

0.00

0.00

CH2OH  O2  CH2O  HO2

2.41  10

0.00

21.00

1.51  10

15

1.00

0.00

4.5

b

14

4.6

CH2OH  HO2  CH2O  H2O2

1.20  10

13

0.00

0.00

4.7

CH2OH  HCO  CH2O  CH2O

1.50  10

13

0.00

0.00

8.30  1017

1.20

64.85

CH3O Reactions 4.8

CH3O  M  CH2O  H  M

(continues)

664

Combustion

TABLE C4 (continued) A

n

E

4.9

CH3O  O  CH2O  OH

6.00  10

12

4.10

CH3O  OH  CH2O  H2O

1.80  10

13

CH3O  O2  CH2O  HO2

9.03  10

0.00

50.12

2.20  10

0.00

7.31

11

3.00  10

0.00

0.00

CH2OH  H  CH3OH, k

1.06  1012

CH2OH  H  M CH3OH  M, k0

4.36  10

b

4.11

13 10

4.12

CH3O  HO2  CH2O  H2O2

0.00

0.00

0.00

0.00

CH3OH Reactions 4.13c,d

0.50

0.36

31

4.65

21.26

CH3O  H  CH3OH, k

2.43  1012

0.50

0.21

CH3O  H  M  CH3OH  M, k0

4.66  10

7.44

58.91

α  0.6, T  1.0  10 , T  9.0  10 , T**  1  104 ***

2

*

4

εH2  2, εH2O  6, εCO  1.5, εCO2  2, εCH4  2 4.14c,d

41

α  0.7, T  1.0  10 , T  9.0  10 , T**  1  104 ***

2

*

4

εH2  2, εH2O  6, εCO  1.5, εCO2  2, εCH4  2 4.15

CH3OH  H  CH2OH  H2

3.20  1013

0.00

25.50

4.16

CH3OH  H  CH3O  H2

8.00  1012

0.00

25.50

4.17

CH3OH  O  CH2OH  OH

3.88  10

2.50

12.89

4.18

CH3OH  OH  CH3O  H2O

1.00  10

2.10

2.08

4.19

CH3OH  OH  CH2OH  H2O

7.10  10

1.80

2.49

4.20

CH3OH  O2  CH2OH  HO2

2.05  1013

0.00

187.86

4.21

CH3OH  HO2  CH2OH  H2O2

3.98  10

13

0.00

81.17

4.22

CH2OH  HCO  CH3OH  CO

1.00  1013

0.00

0.00

4.23

CH3O  HCO  CH3OH  CO

9.00  10

13

0.00

0.00

4.24

CH3OH  HCO  CH2OH  CH2O

9.64  10

3

2.90

54.85

5 6 6

665

Appendix C: Specific Reaction Rate Constants

TABLE C4 (continued) A

n

E

4.25

CH2OH  CH2OH  CH3OH  CH2O

3.00  10

0.00

0.00

4.26

CH3O  CH2OH  CH3OH  CH2O

2.40  1013

0.00

0.00

4.27

CH3O  CH3O  CH3OH  CH2O

13

6.00  10

0.00

0.00

4.28

CH3OH  CH3O  CH3OH  CH2OH

3.00  1011

0.00

16.99

12

Source: Li, J., Zhao, Z., Kazakov, A., Chaos, M., Dryer, F. L., and Scire, J. J., Jr., A Comprehensive Kinetic Mechanism for CO, CH2O, and CH3OH Combustion, Int. J. Chem. Kinet. 39, 109–136 (2007). a Reaction rates in cm3 mol s kJ units, k  ATn exp(E/RT). b Rate represented by sum of two Arrhenius expressions, using the Arrhenius parameters of the current reaction line and the line below. c ki  εi  kM, εi  1 for chemical species not defined. d The fall-off behavior of this reaction is expressed as k  k[Pr/(1  Pr)]F, Pr  k0[M]/ k, log(F)  [1  [(log Pr  c)/(n  d{log Pr  c})]2]1 log Fcent, c  0.4  0.67log Fcent, n  0.75  1.27 log Fcent, d  0.14, and Fcent  (1  α)exp(T/T***  α exp(T/T*)  exp(T/T**), Gilbert, R. G., Luther, K., and Troe, J., Ber. Bunsenges. Phys. Chem. 87, 169 (1983). e Because of the formation of CH3 radicals, the methanol oxidation mechanism of Li et al. (the source above) also includes reactions 5.30–5.33, 5.36–5.47, 5.49, and 6.82. The heat of formation for OH used by Li et al. was ΔHf,298.15(OH)  8.91 kcal/mol [Ruscic, B., Wagner A.F., Harding, L.B., Asher, R.L., Feller, D., Dixon, D.A., Peterson, K.A., Song, Y., Qian, X.M., Ng, C.Y., Liu, J.B., and Chen, W.W., J. Phys. Chem. A. 106, 2727 (2002)] and thermochemical data for CH2OH was from Johnson R.O., and Hudgens J.W., Journal of Physical Chemistry100:19874, 1996). See the source above for more details.

TABLE C5 CH4/CH3OH/CH2O/CO/H2/O2 Mechanisma A

n

E

C Reactions 5.1

C  OH  CO  H

5.00  1013

0.00

0.00

5.2

C  O2  CO  O

2.00  10

13

0.00

0.00

CH Reactions 5.3

CH  H  C  H2

1.50  1014

0.00

0.00

5.4

CH  O  CO  H

5.70  10

13

0.00

0.00

5.5

CH  OH  HCO  H

3.00  10

13

0.00

0.00

5.6

CH  O2  HCO  O

3.30  10

13

0.00

0.00

5.7

CH  H2O  CH2O  H

1.17  10

15

0.75

0.00

5.8

CH  CO2  HCO  CO

3.40  10

12

0.00

2.88

1.56

0.00

0.00

0.00

CH2 Reactions 5.9

CH2  H  CH  H2

1.00  1018

5.10

CH2  O  CO  H  H

5.00  10

13

(continues)

666

Combustion

TABLE C5 (continued) A

n

E

5.11

CH2  O  CO  H2

3.00  10

0.00

0.00

5.12

CH2  OH  CH2O  H

2.50  10

13

0.00

0.00

5.13

CH2  OH  CH  H2O

1.13  10

7

5.14

CH2  O2  CO  H  OH

5.15

13

2.00

12.55

10

8.60  10

0.00

2.10

CH2  O2  CO  H2O

1.90  10

10

0.00

4.18

5.16

CH2  O2  CO2  H  H

1.60  10

0.00

4.18

5.17

CH2  O2  CO2  H2

6.90  10

0.00

2.10

5.18

CH2  O2  HCO  OH

4.30  10

0.00

2.10

5.19

CH2  O2  CH2O  O

5.00  10

0.00

37.65

5.20

CH2  HO2  CH2O  OH

1.81  10

0.00

0.00

5.21

CH2  CO2  CH2O  CO

11

1.10  10

0.00

4.18

1.00  1013

0.00

0.00

13 11 10 13 13

1

CH2 Reactions

5.22b

1

CH2  M  CH2  M

εH  0 5.23

1

CH2  H  CH2  H

2.00  1014

0.00

0.00

5.24

1

CH2  O  CO  H2

1.51  1013

0.00

0.00

5.25

1

CH2  O  HCO  H

1.51  1013

0.00

0.00

5.26

1

CH2  OH  CH2O  H

3.00  1013

0.00

0.00

5.27

1

CH2  O2  CO  OH  H

13

3.00  10

0.00

0.00

9.00  1013

0.00

63.18

CH3 Reactions 5.28

CH3  H  CH2  H2

5.29

1

CH2  H2  CH3  H

7.00  10

0.00

0.00

5.30

CH3  O  CH2O  H

8.43  10

13

0.00

0.00

5.31

CH2OH  H  CH3  OH

9.64  10

13

0.00

0.00

5.32

CH3O  H  CH3  OH

3.20  10

13

0.00

0.00

13

667

Appendix C: Specific Reaction Rate Constants

TABLE C5 (continued) A b,c

5.33

n

CH3  OH  CH3OH, k

2.79  10

18

CH3  OH  M  CH3OH  M, k0

4.00  10

36

E

1.40

5.56

5.92

13.14

2.00

20.92

1.80

33.75 122.30

α  0.412, T  1.95  10 , T*  5.9  103, T**  6.39  103 ***

2

εH2  2, εH2O  6, εCO  1.5, εCO2  2, εCH4  2 5.34

CH3  OH  CH2  H2O

7.50  106

5.35

CH3  OH  CH2  H2O

8.90  10

5.36

CH3  O2  CH3O  O

1.99  10

1.57

5.37

CH3  O2  CH2O  OH

3.74  10

11

0.00

61.26

5.38

CH3  HO2  CH3O  OH

2.41  10

10

0.76

9.73

5.39

CH3O  CO  CH3  CO2

1.60  10

13

0.00

49.37

CH3  H  CH4, k

1.27  1016

0.63

1.60

CH3  HM  CH4  M, k0

2.48  10

4.76

10.21

1

19 18

CH4 Reactions 5.40b,c

33

α  0.412, T  1.95  10 , T*  5.9  103, T**  6.39  103 ***

2

εH2  2, εH2O  6, εCO  1.5, εCO2  2, εCH4  2 5.41

CH4  H  CH3  H2

5.47  107

1.97

46.90

5.42

CH4  O  CH3  OH

3.15  10

12

0.50

43.06

5.43

CH4  OH  CH3  H2O

5.72  10

6

1.96

11.04

5.44

CH3  HO2  CH4  O2

3.16  10

12

5.45

CH4  HO2  CH3  H2O2

5.46

CH3  HCO  CH4  CO

0.00

0.00

1.81  10

11

0.00

77.74

1.20  10

14

0.00

0.00 (continues)

668

Combustion

TABLE C5 (continued) A

n 6

E

5.47

CH3  CH2O  CH4  HCO

3.64  10

5.42

4.18

5.48

CH3  CH3O  CH4  CH2O

2.41  10

0.00

0.00

5.49

CH3  CH3OH  CH4  CH2OH

3.19  10

3.17

30.01

5.50

CH4  CH2  CH3  CH3

4.00  10

0.00

0.00

1

13 1 14

Sources: Miller, J. A., and Bowman, C. T., Mechanism and Modeling of Nitrogen Chemistry in Combustion, Prog. Energy Combust. Sci. 15:287–338 (1989); Li, J., Zhao, Z., Kazakov, A., Chaos, M., Dryer, F. L., and Scire, J. J., Jr., A Comprehensive Kinetic Mechanism for CO, CH2O, and CH3OH Combustion, Int. J. Chem. Kinet. 39, 109–136 (2007); Held, T., The Oxidation of Methanol, Isobutene, and Methyl tertiary-Butyl Ether, No. 1978-T, Ph.D. Dissertation, Princeton University, Princeton, NJ, 1993; Burgess, D. R. F., Jr., Zachariah, M. R., Tsang, W., and Westmoreland, P. R., Thermochemical and Chemical Kinetic Data for Fluorinated Hydrocarbons, NIST Technical Note 1412, NIST, Gaithersburg, MD, 1995. The CH4 and C2H6 (Table C6) mechanisms are based on the high temperature Miller–Bowman mechanism updated and extended to intermediate temperatures with kinetic data from the other listed sources. The reaction sets provided here have not been evaluated against experimental data. For validated CH4 mechanisms, the reader is referred to those given in the text of this appendix. For example, the GRIMECH (Bowman, C. T., Frenklach, M., Gardiner, W., Golden, D., Lissianski, V., Smith, G., and Wang, H., Gas Research Institute Report, 1995) mechanisms are optimized mechanisms for describing the combustion kinetics of methane/air mixtures with nitric oxide chemistry. a Reaction rates in cm3 mol s kJ units, k  ATn exp(E/RT). b ki  εi  kM, εi  1 for chemical species not defined. c The fall-off behavior of this reaction is expressed as k  k[Pr/(1  Pr)]F, Pr  k0[M]/ k, log(F)  [1  [(log Pr  c)/(n  d{log Pr  c})]2]1 log Fcent, c  0.4  0.67 log Fcent, n  0.75  1.27 log Fcent, d  0.14, and Fcent  (1  α)exp(T/T***  αexp(T/T*)  exp(T/T**), Gilbert, R. G., Luther, K., and Troe, J., Ber. Bunsenges. Phys. Chem. 87, 169 (1983).

TABLE C6 C2H6/CH4/CH3OH/CH2O/CO/H2/O2 Mechanisma A

n

E

C2H Reactions 6.1

CH2  C  C2H  H

5.00  1013

0.00

0.00

6.2

C2H  O  CH  CO

13

5.00  10

0.00

0.00

6.3

C2H  O2  HCO  CO

2.41  1012

0.00

0.00

HCCO Reactions 6.4

HCCO  H  1CH2  CO

1.00  1014

0.00

0.00

6.5

C2H  OH  HCCO  H

2.00  1013

0.00

0.00

6.6

HCCO  O  H  CO  CO

1.00  1014

0.00

0.00

6.7

C2H  O2  HCCO  O

6.02  1011

0.00

0.00

6.8

HCCO  O2  CO  CO  OH

1.60  10

0.00

3.58

12

669

Appendix C: Specific Reaction Rate Constants

TABLE C6 (continued) A

n

E

3.70

531.78

C2H2 Reactions 6.9

C2H2  M  C2H  H  M

7.46  1030

6.10

CH3  C  C2H2  H

5.00  10

0.00

0.00

6.11

CH2  CH  C2H2  H

4.00  10

0.00

0.00

6.12

C2H  H2  C2H2  H

4.09  10

2.39

3.61

6.13

C2H2  O  CH2  CO

1.02  10

2.00

7.95

6.14

C2H2  O  C2H  OH

3.16  1015

0.60

62.76

6.15

C2H2  O  HCCO  H

1.02  107

2.00

7.95

6.16

C2H2  OH  CH3  CO

4.83  104

4.00

8.37

6.17

C2H2  OH  C2H  H2O

3.38  107

2.00

58.58

6.18

CH2  CH2  C2H2  H2

4.00  1013

0.00

0.00

6.19

C2H2  O2  C2H  HO2

1.20  10

0.00

311.75

6.20

C2H2  O2  HCCO  OH

2.00  10

1.50

125.94

6.21

HCCO  CH  C2H2  CO

5.00  10

0.00

0.00

6.22

HCCO  HCCO  C2H2  2CO

13

1.00  10

0.00

0.00

C2H2  H  C2H3, k

5.54  1012

C2H2  H  M  C2H3  M, k0

2.67  10

13 13 5 7

13 8 13

C2H3 Reactions 6.23b

0.00

10.08

27

3.50

10.08

εH2  2, εCO  2, εH2O  5, εCO2  3 6.24

C2H3  H  C2H2  H2

1.20  1013

0.00

0.00

6.25

CH3  CH  C2H3  H

3.00  10

0.00

0.00

6.26

C2H3  OH  C2H2  H2O

3.00  10

0.00

0.00

6.27

C2H3  O2  HCO  HCO  H

3.27  10

3.94

20.96

C2H3  O2  CH2O  HCO

4.48  10

4.55

22.93

1.05  10

38

8.22

29.41

6.28

c

13 13 23 26

6.29

C2H3  O2  C2H2  HO2

5.10  1021

3.24

23.68

6.30

HCCO  CH2  C2H3  CO

3.00  1013

0.00

0.00 (continues)

670

Combustion

TABLE C6 (continued) A

n

E

6.31

C2H3  CH  C2H2  CH2

5.00  10

13

0.00

0.00

6.32

C2H3  CH3  C2H2  CH4

3.90  10

11

0.00

0.00

6.33

C2H3  C2H3  C2H4  C2H2

9.60  10

11

0.00

0.00

5.04  105

2.30

56.49

CH2CO  CH2  CO, k

3.00  1014

0.00

297.00

CH2CO  M  CH2  CO  M, k0

3.60  10

15

0.00

247.99

6.36

CH2CO  H  CH3  CO

1.13  10

13

0.00

14.34

6.37

CH2CO  H  HCCO  H2

5.00  1013

0.00

33.47

6.38

C2H2  OH  CH2CO  H

2.18  104

4.50

4.18

6.39

C2H3  O  CH2CO  H

3.00  1013

0.00

0.00

6.40

CH  CH2O  CH2CO  H

9.46  1013

0.00

2.15

6.41

HCCOH  H  CH2CO  H

1.00  1013

0.00

0.00

6.42

CH2CO  O  CH2  CO2

1.75  10

12

0.00

5.65

6.43

CH2CO  O  HCCO  OH

1.00  10

13

0.00

33.47

6.44

CH2CO  OH  HCCO  H2O

7.50  10

12

0.00

8.37

6.45

C2H3  HO2  CH2CO  OH  H

3.00  10

13

0.00

0.00

CH3CO  CH3  CO, k

1.20  1022

3.04

78.66

CH3CO  M  CH3  CO  M, k0

8.73  10

42

8.62

93.72

6.47

CH3CO  H  CH3  HCO

9.60  10

13

0.00

0.00

6.48

CH3CO  O  CH3  CO2

9.60  10

12

0.00

0.00

6.49

CH3CO  OH  CH3  CO  OH

3.00  10

13

0.00

0.00

6.50

CH3CO  OH  CH2CO  H2O

1.20  10

13

0.00

0.00

6.51

CH3CO  HO2  CH3  CO2  OH

3.00  10

13

0.00

0.00

C2H4  M  C2H2  H2  M

1.50  1015

HCCOH Reactions 6.34

C2H2  OH  HCCOH  H

CH2CO Reactions 6.35

CH3CO Reactions 6.46

C2H4 Reactions 6.52

0.44

233.48

671

Appendix C: Specific Reaction Rate Constants

TABLE C6 (continued) A

n

E

6.53

C2H4  M  C2H3  H  M

1.40  10

0.00

344.61

6.54

C2H4  H  C2H3  H2

1.10  10

0.00

35.57

6.55

CH4  CH  C2H4  H

6.00  10

0.00

0.00

6.56

CH3  CH2  C2H4  H

3.00  10

0.00

0.00

6.57

C2H4  O  CH3  HCO

1.60  10

1.20

3.12

6.58

C2H4  OH  C2H3  H2O

2.02  1013

0.00

24.92

6.59

C2H4  O2  C2H3  HO2

4.22  1013

0.00

241.00

6.60

C2H4  CH3  C2H3  CH4

6.62

3.70

39.75

6.61

CH3  CH3  C2H4  H2

1.00  1016

0.00

139.91

16 14 13 13 9

C2H4O Reactions 6.62

C2H4O  CH4  CO

3.16  1014

0.00

238.49

6.63

C2H4  HO2  C2H4O  OH

6.00  10

0.00

33.26

9

CH3CHO Reactions 6.64

CH3CHO  CH3  HCO

7.08  1015

0.00

342.10

6.65

C2H3  OH  CH3CHO

3.00  10

0.00

0.00

6.66

CH3CHO  H  CH3CO  H2

4.00  10

0.00

17.60

6.67

CH3CHO  O  CH3CO  OH

5.00  10

0.00

7.50

6.68

CH3CHO  OH  CH3CO  H2O

1.00  10

0.00

0.00

6.69

CH3CHO  O2  CH3CO  HO2

2.00  10

0.50

176.56

6.70

CH3CHO  HO2  CH3CO  H2O2

1.70  10

0.00

44.77

6.71

CH3CHO  CH3  CH3CO  CH4

1.74  10

0.00

35.31

C2H5  C2H4  H, k

4.90  109

1.19

155.64

C2H5  M  C2H4  H  M, k0

5.10  10

14.00

251.46

6.73

CH3  CH3  C2H5  H

15

8.00  10

0.00

110.93

6.74

C2H5  H  C2H4  H2

1.81  1012

0.00

0.00

6.75

C2H5  O  CH2O  CH3

1.60  1013

0.00

0.00

6.76

C2H5  O  CH3HCO  H

9.60  1014

0.00

0.00

13 13 12 13 13 12 12

C2H5 Reactions 6.72

64

(continues)

672

Combustion

TABLE C6 (continued) A

n

E

6.77

C2H5  OH  CH3  CH2O  H

2.40  10

13

0.00

0.00

6.78

C2H5  OH  C2H4  H2O

2.40  10

13

0.00

0.00

6.79

C2H5  O2  C2H4  HO2

2.56  10

19

2.77

8.27

6.80

C2H5  HO2  CH3  CH2O  OH

2.40  10

13

0.00

0.00

6.81

C2H5  HO2  C2H4  H2O2

3.00  10

11

0.00

0.00

CH3  CH3  C2H6, k

9.21  1016

1.17

2.67

CH3  CH3  M  C2H6  M, k0

1.14  1036

5.25

7.13

7.08

445.60

C2H6 Reactions 6.82b,d

Fc  (0.64) exp(T/6927)  0.64 exp(132/T) εH2  2, εH2O  5, εCO  2, εCO2  3 6.83

C2H6  C2H5  H

2.08  1038

6.84

C2H6  H  C2H5  H2

5.42  102

3.50

21.80

6.85

C2H6  O  C2H5  OH

3.00  107

2.00

21.40

6.86

C2H6  OH  C2H5  H2O

5.13  10

6

2.06

3.57

6.87

C2H6  O2  C2H5  HO2

4.00  10

13

0.00

213.00

6.88

C2H6  HO2  C2H5  H2O2

2.94  10

11

0.00

62.51

6.89

C2H6  CH2  C2H5  CH3

1.20  10

14

0.00

0.00

1

6.90

C2H6  CH3  C2H5  CH4

5.50  10

1

4.00

34.73

6.91

C2H6  C2H3  C2H5  C2H4

6.00  10

2

3.30

43.93

6.92

C2H5  C2H5  C2H6  C2H4

1.40  10

12

0.00

0.00

Sources: Miller, J. A., and Bowman, C. T., Mechanism and Modeling of Nitrogen Chemistry in Combustion, Prog. Energy Combust. Sci. 15, 287–338 (1989); Li, J., Zhao, Z., Kazakov, A., Chaos, M., Dryer, F. L., and Scire, J.J., Jr., A Comprehensive Kinetic Mechanism for CO, CH2O, and CH3OH Combustion, Int. J. Chem. Kinet. 39, 109–136 (2007); Held, T., The Oxidation of Methanol, Isobutene, and Methyl tertiary-Butyl Ether, No. 1978-T, Ph.D. Dissertation, Princeton University, Princeton, NJ, 1993; Burgess, D. R. F., Jr., Zachariah, M. R., Tsang, W., and Westmoreland, P. R., Thermochemical and Chemical Kinetic Data for Fluorinated Hydrocarbons, NIST Technical Note 1412, NIST, Gaithersburg, MD, 1995. a Reaction rates in cm3 mol s kJ units, k  ATn exp(E/RT). b ki  εi  kM, εi  1 for chemical species not defined. c Rate represented by sum of two Arrhenius expressions, using the Arrhenius parameters of the current reaction line and the line below. d The fall-off behavior of this reaction is expressed as k  [k0k/(k0  k/M)]  F, and log(F)  log(Fc)/[1  {log(k0  M/k)}2].

673

Appendix C: Specific Reaction Rate Constants

Complete mechanisms for the high temperature oxidation of propane and larger hydrocarbons are available in the literature [e.g., Warnatz, J., Proc. Combust. Inst., 24, 553–579. (1992), and Ranzi, E., Sogaro, A., Gaffuri, P., Pennati, G., Westbrook, C. K., and Pitz, W. J., Combust. Flame, 99, 201 (1994)]. Because of the space limitations, only selected reactions for propane oxidation are presented in Table C7.

TABLE C7 Selected Reactions of a C3H8 Oxidation Mechanisma A

n

E

C3H8 Reactions 7.1

C3H8  C2H5  CH3

7.90  1022

1.80

371.10

1.33  10

6

7.2

C3H8  H  n-C3H7  H2

2.54

28.27

C3H8  H  i-C3H7  H2

1.30  10

6

7.3

2.40

18.71

7.4

C3H8  O  n-C3H7  OH

1.93  10

2.68

15.55

7.5

C3H8  O  i-C3H7  OH

4.76  10

2.71

8.81

7.6

C3H8  OH  n-C3H7  H2O

1.05  10

0.97

13.19

7.7

C3H8  OH  i-C3H7  H2O

4.67  10

1.61

0.29

7.8

C3H8  O2  n-C3H7  HO2

3.98  10

0.00

213.10

7.9

C3H8  O2  i-C3H7  HO2

3.98  10

0.00

199.10

7.10

C3H8  HO2  n-C3H7  H2O2

4.75  10

2.55

69.00

7.11

C3H8  HO2  i-C3H7  H2O2

3

9.64  10

2.60

58.20

1

5 4 10 7 13 13 4

7.12

C3H8  CH3  n-C3H7  CH4

9.03  10

3.65

59.47

7.13

C3H8  CH3  i-C3H7  CH4

1.51

3.46

45.56

n-C3H7 Reactions 7.14

n-C3H7  C2H4  CH3

1.26  1013

0.00

252.75

7.15

n-C3H7  C3H6  H

1.12  1013

0.00

293.23

7.16

n-C3H7  O2  C3H6  HO2

1.17  1019

1.59

112.16

7.17

n-C3H7  O2  C2H4  CH2O  OH

2.44  1016

0.95

178.58

2.00  1010

0.00

245.26

i-C3H7 Reactions 7.18

i-C3H7  C2H4  CH3

(continues)

674

Combustion

TABLE C7 (continued) A

n

E

7.19

i-C3H7  C3H6  H

4.00  10

0.00

317.59

7.20

i-C3H7  O2  C3H6  HO2

27

2.69  10

4.60

117.56

0.00

362.68

13

C3H6 Reactions 7.21

C3H6  C3H5  H

2.50  1015

7.22

C3H6  C2H3  CH3

1.10  10

1.20

408.80

7.23

C3H6  H  C3H5  H2

1.73  10

2.50

20.70

7.24

C3H6  H  CHCHCH3  H2

5

8.07  10

2.50

102.10

7.25

C3H6  H  CH3CCH2  H2

4.10  105

2.50

81.43

7.26

C3H6  O  C3H5  OH

1.75  1011

0.70

48.91

7.27

C3H6  O  CHCHCH3  OH

1.20  1011

0.70

74.49

7.28

C3H6  O  CH3CCH2  OH

6.02  1010

0.70

63.45

7.29

C3H6  O  CH2CO  CH3  H

7.80  107

1.83

3.12

7.30

C3H6  O  CH3CHCO  2H

3.90  10

1.83

3.12

C3H6  O  C2H5  HCO

3.49  10

1.83

2.29

1.17  10

1.83

3.12

7.31

b

21 5

7 7 7

7.32

C3H6  OH  C3H5  H2O

3.12  10

2.00

2.47

7.33

C3H6  OH  CHCHCH3  H2O

2.14  10

2.00

11.62

7.34

C3H6  OH  CH3CCH2  H2O

1.11  10

2.00

6.07

7.35

C3H6  O2  C3H5  HO2

6.02  10

0.00

199.10

7.36

C3H6  HO2  C3H5  H2O2

9.64  10

2.60

58.20

7.37

C3H6  CH3  C3H5  CH4

2.22

3.50

23.74

7.38

C3H6  CH3  CHCHCH3  CH4

1.35

3.50

53.76

8.42  10

3.50

4.88

7.39

C3H6  CH3  CH3CCH2  CH4

6 6 6 13 3

1

CHCHCH3 Reactions 7.40

CHCHCH3  C2H2  CH3

1.59  1012

0.00

156.90

7.41

CHCHCH3  H  C3H4  H2

3.33  1012

0.00

0.00

7.42

CHCHCH3  O2  CH3HCO  HCO

4.34  1012

0.00

0.00

7.43

CHCHCH3  CH3  C3H4  CH4

1.00  1011

0.00

0.00

675

Appendix C: Specific Reaction Rate Constants

TABLE C7 (continued) A

n

E

CH2CCH3 Reactions 7.44

CH2CCH3  H  pC3H4  H2

3.33  1012

0.00

0.00

7.45

CH2CCH3  O2  CH2O  CH3CO

4.34  10

12

0.00

0.00

7.46

CH2CCH3  CH3  pC3H4  CH4

1.00  10

11

0.00

0.00

C3H5 Reactions 7.47

C3H5  C3H4  H

1.40  1013

0.00

251.00

7.48

C3H5  H  C3H4  H2

13

1.80  10

0.00

0.00

7.49

C3H5  O  C2H3HCO  H

6.02  10

13

0.00

0.00

7.50

C3H5  OH  C3H4  H2O

6.02  10

12

0.00

0.00

7.51

C3H5  O2  C3H4  HO2

1.33  107

0.00

0.00

7.52

C3H5  HO2  C2H3HCO  OH  H

1.92  1011

0.00

0.00

7.53

C3H5  CH3  C3H4  CH4

3.00  1012

0.32

1.10

C3H4 Reactions 7.54

C3H4  pC3H4

1.20  1015

0.00

386.60

7.55

C3H4  M  C3H3  H  M

1.14  1017

0.00

292.90

7

7.56

C3H4  H  C3H3  H2

3.36  10

6.00

7.08

7.57

C3H4  H  CH2CCH3

8.50  10

12

0.00

8.37

3.00  10

3

4.61

17.75

4.61

17.75

7.58

C3H4  O  CH2O  C2H2

7.59

C3H4  O  CO  C2H4

9.00  10

3

7.60

C3H4  OH  C3H3  H2O

1.45  10

13

7.61

C3H4  OH  CH2CO  CH3

7.62

0.00

17.45

12

3.12  10

0.00

1.66

C3H4  O2  C3H3  HO2

4.00  10

13

0.00

257.32

7.63

C3H4  HO2  CH2CO  CH2  OH

4.00  10

12

0.00

79.50

7.64

C3H4  CH3  C3H3  CH4

2.00  10

12

0.00

32.22

1.00  1017

0.00

292.88

p-C3H4 Reactions 7.65 7.66

pC3H4  M  C3H3  H  M pC3H4  C2H  CH3

4.20  10

16

0.00

418.40 (continues)

676

Combustion

TABLE C7 (continued) A

n

E

12

7.67

pC3H4  H  C3H3  H2

1.00  10

0.00

6.28

7.68

pC3H4  H  CH2CCH3

6.50  1012

0.00

8.37

7.69

pC3H4  H  CHCHCH3

12

5.80  10

0.00

12.97

7.70

pC3H4  O  C2H2  CO  H2

1.50  1013

0.00

8.83

7.71

pC3H4  O  CH2CO  CH2

12

3.20  10

0.00

8.41

7.72

pC3H4  O  C2H3  HCO

3.20  1012

0.00

8.41

7.73

pC3H4  O  C2H4  CO

3.20  10

0.00

8.41

7.74

pC3H4  O  HCCO  CH3

12

6.30  10

0.00

8.41

7.75

pC3H4  O  HCCO  CH2  H

3.20  1011

0.00

8.41

7.76

pC3H4  OH  C3H3  H2O

1.50  10

7.77

pC3H4  OH  CH2CO  CH3

5.00  104

4.50

4.18

7.78

pC3H4  O2  C3H3  HO2

2.50  1012

0.00

213.38

7.79

pC3H4  O2  HCCO  OH  CH2

1.00  107

1.50

125.94

7.80

pC3H4  HO2  C2H4  CO  OH

12

3.00  10

0.00

79.50

7.81

pC3H4  CH3  C3H3  CH4

2.00  1012

0.00

32.22

12

3

3.00

0.84

c-C3H4 Reactions 7.82

cC3H4  C3H4

1.51  1014

0.00

210.90

7.83

cC3H4  pC3H4

1.20  1015

0.00

182.80

1.20  1013

C3H3 Reactions 7.84

CH2  C2H2  C3H3  H

0.00

27.61

7.85

1

13

3.00  10

0.00

0.00

7.86

C3H3  O  CH2O  C2H

2.00  1013

0.00

0.00

7.87

C3H3  OH  C3H2  H2O

13

2.00  10

0.00

0.00

7.88

C3H3  O2  CH2CO  HCO

3.00  1010

0.00

12.00

7.89

C3H3  CH3  C2H5  C2H

1.00  1013

0.00

156.90

7.90

C3H3  C3H3  C2H2  C2H2  C2H2

11

5.00  10

0.00

0.00

CH2  C2H2  C3H3  H

C3H2 Reactions 7.91

CH  C2H2  C3H2  H

1.00  1014

0.00

0.00

7.92

C3H2  O2  HCCO  HCO

1.00  1013

0.00

0.00

Sources: Tsang, W., J. Phys. Chem. Ref. Data, 17, 887 (1988) and 20, 221 (1991); Pitz, W. J. and Westbrook, C. K., CDAT database for HCT, LNL, Livermore CA., 1995; Dagaut, P., Cathonnet, M., and Boettner, J.-C., Combust. Sci. and Tech., 71, 111 (1990); Miller, J. A., and Bowman, C. T., Mechanism and Modeling of Nitrogen Chemistry in Combustion, Prog. Energy Combust. Sci., 15, 287–338 (1989). a Reaction rates in cm3 mol s kJ units, k  ATn exp(E/RT). b Rate represented by sum of two Arrhenius expressions.

677

Appendix C: Specific Reaction Rate Constants

Significant progress has also been made on the development of low and intermediate temperature hydrocarbon oxidation mechanisms, and the reader is again referred to the literature. Rate constant data for reactions of post-combustion gases including nitrogen oxides, hydrogen chloride, ozone, and sulfur oxides are presented in Tables C8–C11.

TABLE C8 NxOy/CO/H2/O2 Mechanisma A

n

E

N and NO Reactions 8.1

NO  M  N  O  M

9.64  1014

0.00

620.91

8.2

NO  H  N  OH

1.69  1014

0.00

204.18

8.3

NO  O  N  O2

1.81  109

1.00

162.13

8.4

N  HO2  NO  OH

1.00  1013

0.00

8.39

8.5

NO  N  N2  O

3.27  10

12

0.30

0.00

NO  O  NO2, k

1.30  1015

0.75

0.00

NO  O  M  NO2  M, k0

4.72  10

2.87

6.49

NO  O  Ar  NO2  Ar, k0

19

7.56  10

1.41

0.00

NO2 Reactions 8.6b,d

4

Fc  0.95  1.0  10

24

T

8.7

NO2  H  NO  OH

1.32  1014

0.00

1.51

8.8

NO2  O  NO  O2

12

3.91  10

0.00

1.00

8.9

NO2  OH  NO  HO2

1.81  10

13

0.00

27.93

8.10

NO2  CO  NO  CO2

9.03  10

13

0.00

141.34

8.11

NO2  HCO  NO  H  CO2

8.39  10

15

0.75

8.06

8.12

NO2  N  N2  O2

1.00  10

12

8.13

NO2  N  NO  NO

8.14

NO2  NO2  2NO  O2

0.00

0.00

12

4.00  10

0.00

0.00

1.63  10

12

0.00

109.29

N2O  N2  O, k

7.91  1010

0.00

234.39

N2O  M  N2  O  M, k0

9.13  10

0.00

241.42

N2O Reactions 8.15b

14

εAr  0.63, εH2O  7.5 (continues)

678

Combustion

TABLE C8 (continued) A c

8.16

N2O  H  N2  OH

n

E

2.53  10

0.00

19.04

2.23  10

0.00

70.08

10 14

8.17

N2O  O  N2  O2

1.00  10

0.00

117.15

8.18

N2O  O  NO  NO

1.00  10

0.00

117.15

8.19

NO2  N  N2O  O

5.00  10

0.00

0.00

8.20

N2O  OH  N2  HO2

2.00  10

0.00

167.36

8.21

N2O  CO  N2  CO2

13

5.01  10

0.00

184.10

8.22

N2O  N  N2  NO

1.00  1013

0.00

83.14

8.23

N2O  NO  N2  NO2

1.00  1014

0.00

209.20

NO  H  HNO, k

1.52  1015

0.41

0.00

NO  H  M  HNO  M, k0

8.96  1019

1.32

3.08

14 14 12 12

HNO Reactions 8.24b,d

Fc  0.82 εAr  0.63 8.25

HNO  H  NO  H2

4.46  1011

0.72

2.74

8.26

HNO  O  NO  OH

1.81  1013

0.00

0.00

8.27

HNO  OH  NO  H2O

1.30  10

7

1.88

4.00

8.28

NO  HCO  HNO  CO

7.23  10

12

0.00

0.00

8.29

HNO  HCO  NO  CH2O

6.02  10

11

8.30

HNO  N  N2O  H

8.31 8.32

0.00

8.31

10

5.00  10

0.50

12.55

HNO  NO  N2O  OH

2.00  10

12

0.00

08.78

HNO  HNO  N2O  H2O

8.51  10

8

0.00

12.89

HONO Reactions 8.33b,dc

NO  OH  HONO, k

1.99  1012

0.05

3.02

NO  OH  M  HONO  M, k0

5.08  10

23

2.51

0.28

3.21  1012

0.00

120.54

Fc  0.62 εAr  0.63 8.34

NO2  H2  HONO  H

679

Appendix C: Specific Reaction Rate Constants

TABLE C8 (continued) A

n

E

13 10

8.35

HONO  O  OH  NO2

1.20  10

0.00

24.94

8.36

HONO  OH  H2O  NO2

1.26  10

1.00

0.57

8.37

NO2  HCO  HONO  CO

1.24  10

7.29

9.85

8.38

NO2  CH2O  HONO  HCO

7.83  10

2.77

57.45

8.39

HNO  NO2  HONO  NO

6.02  1011

23 2

0.00

8.31

NH Reactions 8.40

NH  M  N  H  M

2.65  1014

0.00

315.93

8.41

N  HO2  NH  O2

1.00  10

0.00

8.37

8.42

NH  O2  NO  OH

7.60  10

0.00

6.40

8.43

NH  O2  HNO  O

13

3.89  10

0.00

74.85

8.44

N  H2  NH  H

1.60  10

14

0.00

105.19

8.45

NH  O  N  OH

3.72  10

0.00

0.00

8.46

NH  O  NO  H

5.50  10

0.00

0.00

8.47

NH  OH  N  H2O

11

5.00  10

0.50

8.37

8.48

NH  OH  HNO  H

2.00  1013

0.00

0.00

8.49

NH  N  N2  H

3.00  1013

0.00

0.00

8.50

NH  NO  N2  OH

2.16  1013

0.23

0.00

8.51c

NH  NO  N2O  H

2.94  1014

0.40

0.00

2.16  1013

0.23

0.00

13 10

13 13

8.52

HNO  N  NH  NO

1.00  10

0.00

8.37

8.53

NH  NO2  HNO  NO

1.00  10

0.50

16.74

8.54

NH  NH  N2  HH

13

5.10  10

0.00

0.00

2.00

382.44

13 11

NH2 Reactions 8.55

NH2  M  NH  H  M

3.98  1023

8.56

NH2  H  NH  H2

7.20  10

2.32

6.65

8.57

NH2  O  HNO  H

6.63  10

0.50

0.00

8.58

NH2  O  NH  OH

12

6.75  10

0.00

0.00

8.59

NH2  OH  NH  H2O

4.00  10

12

2.00

4.18

5 14

(continues)

680

Combustion

TABLE C8 (continued) A 8.60 8.61

c

n

E

0.00

62.34

NH2  O2  HNO  OH

1.78  10

NH2  NO  N2  H2O

1.30  10

1.25

0.00

2.80  10

0.55

0.00

12 16 13

8.62

NH2  NO  N2O  H2

5.00  10

0.00

103.09

8.63

NH2  NO  HNO  NH

13

1.00  10

0.00

167.36

8.64

NH2  NO2  N2O  H2O

2.84  1018

2.20

0.00

13

NH3 Reactions 8.65

NH3  M  NH2  H  M

2.20  1016

0.00

391.08

8.66

NH3  H  NH2  H2

6.38  105

2.39

42.68

8.67

NH3  O  NH2  OH

9.40  106

1.94

27.03

8.68

NH3  OH  NH2  H2O

2.04  106

2.04

2.37

8.69

NH2  HO2  NH3  O2

3.00  10

0.00

92.05

8.70

NH2  NH2  NH3  NH

13

5.00  10

0.00

41.84

11

NNH Reactions 8.71

NNH  M  N2  H  M

1.00  1014

0.00

12.47

8.72

NNH  H  N2  H2

1.00  10

0.00

0.00

8.73

NNH  OH  N2  H2O

5.00  10

0.00

0.00

8.74

NH2  NO  NNH  OH

2.80  10

0.55

0.00

8.75

NNH  NO  HNO  N2

5.00  10

0.00

0.00

8.76

NNH  NH  N2  NH2

5.00  10

0.00

0.00

8.77

NNH  NH2  N2  NH3

13

5.00  10

0.00

0.00

14 13 13 13 13

N2H2 Reactions 8.78

N2H2  M  NNH  H  M

1.00  1016

0.00

207.94

8.79

N2H2  M  NH  NH  M

3.16  10

16

0.00

415.89

8.80

N2H2  H  NNH  H2

1.00  1013

0.00

4.16

8.81

NH  NH2  N2H2  H

3.16  1013

0.00

4.16

8.82

N2H2  O  NNH  OH

1.00  1011

0.50

0.00

8.83

N2H2  OH  NNH  H2O

1.00  1013

0.00

8.33

8.84

NH2  NH2  N2H2  H2

3.98  1013

0.00

49.79

681

Appendix C: Specific Reaction Rate Constants

TABLE C8 (continued) A

n

E

8.85

N2H2  HO2  NNH  H2O2

1.00  10

0.00

8.33

8.86

NNH  NNH  N2H2  N2

1.00  10

13

0.00

41.59

8.87

N2H2  NH  NNH  NH2

1.00  10

13

0.00

4.16

8.88

N2H2  NH2  NNH  NH3

1.00  1013

0.00

16.61

13

N2H3 Reactions 8.89

N2H3  M  N2H2  HM

1.00  1016

0.00

207.94

8.90

N2H3  M  NH2  NH  M

1.00  1016

0.00

174.47

8.91

N2H3  H  NH2  NH2

1.58  1012

0.00

0.00

8.92

N2H3  H  NH  NH3

1.00  1011

0.00

0.00

8.93

N2H3  H  N2H2  H2

1.00  1012

0.00

8.33

8.94

N2H3  O  N2H2  OH

11

3.16  10

0.5

0.00

8.95

N2H3  O  NNH  H2O

3.16  10

11

8.96

N2H3  OH  N2H2  H2O

8.97

0.50

0.00

13

1.00  10

0.00

8.33

N2H3  HO2  N2H2  H2O2

1.00  10

13

0.00

8.33

8.98

NH3  NH2  N2H3  H2

7.94  10

0.50

90.37

8.99

N2H2  NH2  NH  N2H3

1.00  10

0.50

141.42

8.100

N2H3  NH2  N2H2  NH3

1.00  10

0.50

0.00

8.101

N2H2  N2H2  NNH  N2H3

13

1.00  10

0.00

41.59

11 11 11

N2H4 Reactions 8.102

N2H4  M  NH2  NH2  M

4.00  1015

0.00

171.13

8.103

N2H4  M  N2H3  H  M

15

1.00  10

0.00

266.10

8.104

N2H4  H  N2H3  H2

1.29  1013

0.00

10.46

8.105

N2H4  H  NH2  NH3

4.46  109

0.00

12.97

8.106

N2H4  O  N2H2  H2O

6.31  1013

0.00

4.98

8.107

N2H4  O  N2H3  OH

2.51  1012

0.00

4.98

8.108

N2H4  OH  N2H3  H2O

3.98  1013

0.00

0.00

8.109

N2H4  HO2  N2H3  H2O2

3.98  1013

0.00

8.33

8.110

N2H4  NH  NH2  N2H3

1.00  10

0.00

8.33

8.111

N2H4  NH2  N2H3  NH3

3.98  10

0.50

8.33

12 11

(continues)

682

Combustion

TABLE C8 (continued) A

n

E

8.112

N2H3  N2H2  N2H4  NNH

1.00  10

13

0.00

41.59

8.113

N2H4  N2H2  N2H3  N2H3

2.50  10

10

0.50

124.68

NO2  O  NO3, k

1.33  1013

0.00

0.00

NO2  OM  NO3  M, k0

1.49  10

4.08

10.32

NO3 Reactions 8.114b,d

4

Fc  0.79  1.8  10

28

T

εAr  0.63 8.115

NO2  NO2  NO3  NO

9.64  109

0.73

87.53

8.116

NO3  H  NO2  OH

6.00  10

0.00

0.00

8.117

NO3  O  NO2  O2

13

1.00  10

0.00

0.00

8.118

NO3  OH  NO2  HO2

1.40  1013

0.00

0.00

8.119

NO3  HO2  NO2  O2  OH

1.50  1012

0.00

0.00

8.120

NO3  NO2  NO  NO2  O2

5.00  1010

0.00

12.30

NO2  OH  HNO3, k

2.41  1013

0.00

0.00

NO2  OH  M  HNO3  M, k0

6.42  1032

5.49

9.83

13

HNO3 Reactions 8.121b,d

Fc  0.725  2.5  104 T εAr  0.63 8.122

NO  HO2  M  HNO3  M

2.23  1012

3.50

9.20

8.123

HNO3  OH  NO2  HO2

1.03  1010

0.00

5.19

8.124

NO3  HO2  HNO3  O2

5.60  10

0.00

0.00

11

Sources: Allen, M. T., Yetter, R. A., and Dryer, F. L., The Decomposition of Nitrous Oxide at 1.5  p  10.5 atm and 1103  T  1173 K, Int. J. Chem. Kinet. 27, 883–909 (1995); Allen, M. T., Yetter, R. A., and Dryer, F. L., High Pressure Studies of Moist Carbon Monoxide/Nitrous Oxide Kinetics, Combustion and Flame, 109,. 449–470 (1997). a Reaction rates in cm3 mol s kJ units, k  ATn exp(E/RT). b ki  εi  kM, εi  1 for chemical species not defined. c Rate represented by sum of two Arrhenius expressions. d The fall-off behavior of this reaction is expressed as k  [k0k/(k0  k/M)]  F, and log(F)  log(Fc)/[1  {log(k0  M/k)}2].

683

Appendix C: Specific Reaction Rate Constants

TABLE C9 HCl/NxOy/CO/H2/O2 Mechanisma A

n

E

HCl and Cl Reactions 9.1

Cl  H  M  HCl  M

7.20  1021

2.00

0.00

9.2

Cl  HO2  HCl  O2

1.08  10

0.00

1.38

9.3

HCl  H  Cl  H2

9.4

HCl  O  Cl  OH

1.69  10

0.30

17.32

3.37  103

2.87

14.67

13 13

9.5

HCl  OH  Cl  H2O

2.71  10

1.65

0.93

9.6

Cl  H2O2  HCl  HO2

6.62  1012

0.00

8.16

9.7

Cl  HCO  HCl  CO

14

1.00  10

0.00

0.00

9.8

Cl  HNO  HCl  NO

9.00  1013

0.00

4.16

9.9

Cl  HONO  HCl  NO2

13

5.00  10

0.00

0.00

9.10

Cl  Cl  M  Cl2  M

4.68  1014

0.00

7.53

9.11

Cl2  H  Cl  HCl

8.59  10

13

0.00

4.90

7

Cl2 Reactions

ClO Reactions 9.12

ClO  O  Cl  O2

5.70  1013

0.00

1.52

9.13

Cl  HO2  ClO  OH

2.42  1013

0.00

9.62

9.14

ClO  CO  Cl  CO2

6.03  10

11

0.00

30.96

9.15

Cl2  O  ClO  Cl

2.52  1012

0.00

11.38

9.16

ClO  NO  Cl  NO2

3.85  10

12

0.00

0.59

HOCl Reactions 9.17

HOCl  Cl  OH

1.76  1020

3.01

237.32

9.18

HOCl  ClO  H

8.12  1014

2.09

392.00

9.19

HOCl  HCl  OH

9.55  10

13

0.00

31.88

9.20

ClO  H2  HOCl  H

6.03  10

11

0.00

59.00

9.21

HOCl  O  ClO  OH

6.03  1012

0.00

18.28

9.22

HOCl  OH  ClO  H2O

1.81  10

12

0.00

4.14

9.23

HCO  ClO  HOCl  CO

3.16  1013

0.00

0.00

9.24

HOCl  Cl  Cl2  OH

12

1.81  10

0.00

1.09

9.25

HOCl  Cl  ClO  HCl

7.62  1012

0.00

0.75

CClO Reactions 9.26

COCl  M  Cl  CO  M

1.30  1014

0.00

33.47

9.27

COCl  O2  ClO  CO2

10

7.94  10

0.00

13.81

9.28

COCl  H  HCl  CO

1.00  1014

0.00

0.00

9.29

COCl  O  ClO  CO

1.00  10

0.00

14

0.00 (continues)

TABLE C9 (continued) A

n

E

9.30

COCl  O  Cl  CO2

1.00  10

0.00

0.00

9.31

COCl  OH  HOCl  CO

12

3.30  10

0.00

0.00

9.32

COCl  Cl  Cl2  CO

4.00  1014

0.00

3.35

2.51  1015

0.00

133.47

13

NOCl Reactions NOCl  M  Cl  NO  M

9.33b

εH2 1.6, εCO2  3.5, εNO 1.38 9.34

NOCl  H  HCl  NO

4.60  1013

0.00

3.73

9.35

NOCl  O  ClO  NO

5.00  1012

0.00

12.55

9.36

NOCl  Cl  Cl2  NO

2.41  1013

0.00

0.00

Source: Roesler, J. F., Yetter, R. A., and Dryer, F. L., Kinetic Interactions of CO, NOx, and HCl Emissions in Postcombustion Gases, Combustion and Flame, 100, 495–504 (1995). a Reaction rates in cm3 mol s kJ units, k  ATn exp(E/RT). b ki  εi  kM, εi  1 for chemical species not defined.

TABLE C10 O3/NxOy/CO/H2/O2 Mechanisma A

n

E

O3 Reactions 10.1

O  O2  O3, k

1.69  1012

O  O2  M  O3  M, k0

0.00

0.00

1.78  10

21

2.80

0.00

13

10.2

O3  H  O2  OH

8.43  10

0.00

3.91

10.3

O3  O  O2  O2

4.81  1012

0.00

17.13

10.4

O3  OH  O2  HO2

1.15  1012

0.00

8.31

10.5

O3  H2O  O2  H2O2

6.20  10

1

0.00

0.00

10.6

O3  HO2  2O2  OH

8.43  10

9

0.00

4.99

10.7

O3  CO  O2  CO2

6.02  10

2

0.00

0.00

10.8

O3  HCO  O2  HCO2

5.00  1011 (@ 298 K)

10.9

O3  N  O2  NO

6.00  107 (@ 298 K)

10.10

O3  NO  O2  NO2

1.08  1012

0.00

11.39

10.11

O3  NO2  O2  NO3

7.22  10

0.00

20.37

10

Sources: Atkinson, R., Baulch, D. L., Cox, R. A., Hampson, R. F., Jr., Kerr, J. A., and Troe, J. Evaluated Kinetic and Photochemical Data for Atmospheric Chemistry. Supplement IV. IUPAC Subcommittee on Gas Kinetic Data Evaluation for Atmospheric Chemistry, J. Phys. Chem. Ref. Data 21(6), 1125–1568 (1992). Reaction rates in cm3 mol s kJ units, k  ATn exp(E/RT).

a

TABLE C11 SOx/NxOy/CO/H2/O2 Mechanisma A

n

E

4.00  1014

0.00

448.96

S and SO Reactions 11.1b

SO  M  S  O  M εN2  1.5, εSO2  10, εH2O  10

11.2

S  OH  SO  H

4.00  1013

0.00

0.00

11.3

S  O2  SO  O

5.20  10

1.81

4.99

6

SO2 Reactions 11.4b,c

SO  O  SO2, k

3.20  1013

0.00

0.00

SO  O  M  SO2  M, k0

1.21  1021

1.54

0.00

1.35

0.00

α  0.8, T***  1.0  1030, T*  1.0  1030 !N2  1.5, !SO2  10, !H2O  10 11.5

SO  OH  SO2  H

1.10  1017

11.6

SO  O2  SO2  O

7.60  10

2.37

12.47

11.7

SO2  CO  SO  CO2

12

2.70  10

0.00

202.03

11.8

SO  NO2  SO2  NO

8.40  1012

0.00

0.00

11.9

SO  SO  SO2  S

2.00  1012

0.00

16.63

SO2  O  SO3, k

9.20  1010

0.00

9.98

SO2  OM  SO3  M, k0

4.00  10

4.00

21.97

3

SO3 Reactions 11.10b

28

εSO2  10, εH2O  10 11.11

SO2  OH  SO3  H

4.90  102

2.69

99.58

11.12

SO3  O  SO2  O2

2.00  10

0.00

83.14

11.13

SO2  NO2  SO3  NO

6.30  10

0.00

112.97

11.14

SO3  SO  SO2  SO2

12

1.00  10

0.00

41.57

12 12

SH Reactions 11.15

SH  O2  SO  OH

1.90  1013

0.00

74.83

11.16

S  H2  SH  H

1.40  1014

0.00

80.65

11.17

SH  O  SO  H

1.00  1014

0.00

0.00

11.18

SH  OH  S  H2O

1.00  10

0.00

0.00

13

(continues)

686

Combustion

TABLE C11 (continued) A

n

E

H2S Reactions 11.19b

H2S  M  S  H2  M

1.60  1024

2.61

372.50

εN2  1.5, εSO2  10, εH2O  10 11.20

H2S  H  SH  H2

1.20  107

2.10

2.93

11.21

H2S  O  SH  OH

7.50  107

1.75

12.14

11.22

H2S  OH  SH  H2O

2.70  1012

0.00

0.00

11.23

H2S  S  SH  SH

8.30  1013

0.00

30.76

5.00  1015

0.00

0.00

HSO Reactions 11.24b

SO  H  M  HSO  M εN2  1.5, εSO2  10, εH2O  10

11.25

HSO  O2  SO2  OH

1.00  1012

0.00

41.84

11.26

HSO  H  SH  OH

4.90  1019

1.86

6.53

11.27

HSO  H  S  H2O

1.60  109

1.37

1.42

11.28

HSO  H  H2S  O

1.10  106

1.03

43.51

11.29

HSO  H  SO  H2

1.00  10

0.00

0.00

11.30

HSO  O  SO  OH

1.40  10

0.15

1.26

11.31

HSO  O  SO2  H

4.50  10

0.40

0.00

11.32

HSO  OH  SO  H2O

1.70  10

1.03

1.67

11.33

SH  HO2  HSO  OH

1.00  10

0.00

0.00

4.80  108

1.02

22.34

13 13 14 9 12

HOS Reactions 11.34

HSO  O  HOS  O

HSO2 Reactions 11.35b

HSO2  SO2  H, k

2.00  1011

0.90

76.82

HSO2  M  SO2  HM, k0

3.50  1025

3.29

79.81

1.10  1019

1.73

0.21

εSO2  10, εH2O  10 11.36

HSO  OM  HSO2  M εN2  1.5, εSO2  10, εH2O  10

687

Appendix C: Specific Reaction Rate Constants

TABLE C11 (continued) A

n

E

11.37

HSO2  O2  SO2  HO2

1.00  10

0.00

0.00

11.38

HSO2  H  SO2  H2

3.00  10

0.00

0.00

11.39

HSO2  OH  SO2  H2O

13

1.00  10

0.00

0.00

SO  OH  HOSO, k

1.60  1012

SO  OH  M  HOSO  M, k0

13 13

HOSO Reactions 11.40b

0.50

1.66

27

9.50  10

3.48

4.07

6.90  1019

1.61

6.65

εN2  1.5, εSO2  10, εH2O  10 11.41b

HSO  OM  HOSO  M εN2  1.5, εSO2  10, εH2O  10

11.42

HOSO  M  HOS  O  M

2.50  1030

4.80

498.00

11.43b,c

HOSO  SO2  H, k

1.70  1010

0.80

196.21

HOSO  M  SO2  H  M, k0

1.50  10

4.53

206.19

30

α  0.3, T  1.0  10 T*  1.0  1030 ***

31

,

εSO2  10, εH2O  10 11.44b,c

HOSO  HSO2, k

1.00  109

HOSO  M  HSO2  M, k0

1.70  10

30

α  0.3, T  1.0  10 T*  1.0  1030 ***

1.03

207.85

35

5.64

232.80

1.00  1012

0.00

4.18

,

εSO2  10, εH2O  10 11.45

HOSO  O2  SO2  HO2

10

11.46

HOSO  H  SO  H2O

6.30  10

6.29

7.95

11.47

HOSO  H  SO2  H2

3.00  1013

0.00

0.00

11.48

HSO  OH  HOSO  H

5.30  107

1.57

15.69

11.49

SO2  OH  HOSO  O

3.90  108

1.89

317.98

11.50

SO3  H  HOSO  O

2.50  105

2.92

210.46

11.51

HOSO  OH  SO2  H2O

1.00  10

0.00

0.00

11.52

HSO  NO2  HOSO  NO

5.80  10

0.00

0.00

12 12

(continues)

688

Combustion

TABLE C11 (continued) A

n

E

H2SO Reactions 11.53

HSO  H  H2SO

1.80  1017

2.47

0.21

11.54

H2SO  H2S  O

28

4.90  10

6.66

300.00

HSOH Reactions 11.55

HSO  H  HSOH

2.50  1020

3.14

3.85

11.56

HSOH  SH  OH

39

2.80  10

8.75

314.64

11.57

HSOH  S  H2O

5.80  1029

5.60

228.03

11.58

HSOH  H2S  O

9.80  1016

3.40

361.92

HOSHO Reactions 11.59

HSO  OH  HOSHO

5.20  1028

5.44

13.26

11.60

HOSHO  HOSO  H

6.40  10

5.89

308.78

11.61

HOSHO  H  HOSO  H2

1.00  10

0.00

0.00

11.62

HOSHO  O  HOSO  OH

5.00  10

0.00

0.00

11.63

HOSHO  OH  HOSO  H2O

1.00  10

12

0.00

0.00

SO2  OH  HOSO2, k

7.20  1012

SO2  OH  M  HOSO2  M, k0

30 12 12

HOSO2 Reactions 11.64b,c

0.00

2.99

4.50  10

25

3.30

2.99

α  0.7, T***  1.0  1030, T*  1.0  1030 εN2  1.5, εSO2  10, εH2O  10 11.65

HOSO2  HOSO  O

5.40  1018

2.34

444.76

11.66

HOSO2  SO3  H

1.40  10

2.91

229.70

11.67

HOSO2  O2  SO3  HO2

7.80  10

0.00

2.74

11.68

HOSO2  H  SO2  H2O

1.00  10

0.00

0.00

11.69

HOSO2  O  SO3  OH

5.00  10

0.00

0.00

11.70

HOSO2  OH  SO3  H2O

1.00  10

12

0.00

0.00

11.71

S2  M  S  S  M

4.80  1013

0.00

322.58

11.72

SH  S  S2  H

1.00  1013

0.00

0.00

18 11 12 12

S2 Reactions

689

Appendix C: Specific Reaction Rate Constants

TABLE C11 (continued) A

n

E

11.73

S2  O  SO  S

1.00  10

0.00

0.00

11.74

SH  SH  S2  H2

12

1.00  10

0.00

0.00

1.00  1016

0.00

0.00

13

HS2 Reactions 11.75b

S2  HM  HS2  M εN2  1.5, εSO2  10, εH2O  10

11.76

HS2  H  S2  H2

1.20  107

2.10

2.99

11.77

HS2  O  S2  OH

7.50  10

1.80

12.14

11.78

HS2  OH  S2  H2O

2.70  1012

0.00

0.00

11.79

HS2  S  S2  SH

8.30  1013

0.00

30.76

1.00  1013

0.00

0.00

7

H2S2 Reactions 11.80b

HS2  HM  H2S2  M εN2  1.5, εSO2  10, εH2O  10

11.81

H2S2  H  HS2  H2

1.20  107

2.10

2.99

11.82

H2S2  O  HS2  OH

7.50  10

1.80

12.14

11.83

H2S2  OH  HS2  H2O

2.70  10

0.00

0.00

11.84

H2S2  S  HS2  SH

8.30  10

0.00

30.76

7 12 13

Sources: Dagaut, P., Lecomte, F., Mieritz, J., and Glarborg, P., Experimental Study and Kinetic Modeling Study of the Effect of NO and SO2 on the Oxidation of CO ¶H2 Mixtures, Int. J. Chem. Kinet. 564 (2003); Alzueta, M. A., Bilbao, R., and Glarborg, P., Inhibition and Sensitization of Fuel Oxidation by SO2, Combust. Flame 127, 2234–2252 (2001); Glarborg, P., Kubel, D., Dam-Johnansen, K., Chiang, H-M., and Bozzelli, J., Impact of SOx and NO on CO Oxidation Under Post-Flame Conditions, Int. J. Chem. Kinet. 28, 773–790 (1996). a Reaction rates in cm3 mol s kJ units, k  ATn exp(E/RT). b ki  εi  kM, εi  1 for chemical species not defined. c The fall-off behavior of this reaction is expressed as k  k[Pr/(1  Pr)]F, Pr  k0[M]/ k, log (F)  [1  [(log Pr  c)/(n  d{log Pr  c})]2]1 log Fcent, c   0.4  0.67log Fcent, n  0.75  1.27log Fcent, d  0.14, and Fcent  (1  α)exp(T/T***  α exp(T/T*), Gilbert, R. G., Luther, K., and Troe, J., Ber. Bunsenges. Phys. Chem. 87, 169 (1983).

While Table C8 includes reactions for the formation of thermal NO, it does not include those for prompt NO. Mechanisms and reaction rate data for prompt NO formation and various methods for the reduction of NO have been described by Miller and Bowman [Prog. Energy Combust. Sci. 15, 287(1989)].

690

Combustion

Many detailed reaction mechanisms are available from the Internet. GRI-Mech (www.me.berkeley.edu/gri-mech/) is an optimized detailed chemical reaction mechanism developed for describing methane and natural gas flames and ignition. The last release is GRI-Mech 3.0, which was preceded by versions 1.2 and 2.11. The conditions for which GRI-Mech was optimized are roughly 1000–2500 K, 10 Torr to 10 atm, and equivalence ratios from 0.1 to 5 for premixed systems. The combustion chemistry group at Lawrence Livermore National Laboratory (LLNL) has available detailed mechanisms for hydrogen and various hydrocarbons (e.g., ethanol, dimethyl ether, dimethyl carbonate, CH4, C2H4, C2H6, C3H6, C3H8, nC4H10, methyl butanoate, methyl formate, heptane, iso-octane, and cyclohexane) (http://www-cmls.llnl.gov/?url  science_ and_technology-chemistry-combustion). Other examples include: the Leeds University and Eötvös University CH4/SOx/NOx mechanisms (http://www. chem.leeds.ac.uk/Combustion/Combustion.html), the University of Galway Combustion Chemistry Centre mechanisms (http://www.nuigalway.ie/chem/ c3/mechanisms.htm), the Princeton University mechanisms (F.L. Dryer) (http://www.princeton.edu/~combust/database/other.html), the University of California-San Diego mechanisms (http://www-mae.uscd.edu/~combustion/ cermech/), the Vrije Universiteit Brussel mechanisms (http://homepages.vub. ac.be/~akonnov/science/mechanism/main.html#perform), and the University of Southern California Combustion Kinetics Laboratory mechanisms (http:// ignis.usc.edu/model_release.html#top). The NIST Chemical Kinetics Model Database web site (http://kinetics.nist. gov/CKMech/) is a good resource for chemical kinetic models, thermochemical property data, and elementary rate coefficients. The book Gas-Phase Combustion Chemistry edited by W. C. Gardiner, Jr. (Springer-Verlag, NY, 1999) also lists many detailed mechanisms for different fuels that are available in technical papers and from the Internet. Critical reviews of reaction rate data are constantly appearing in the literature and are an important source for mechanism construction. Some examples of recent reviews are given below. 1. “Evaluated Kinetic Data for Combustion Modeling: Supplement II,” Baulch, D. L., Bowman, C. T., Cobos, C. J., Cox, R. A., Just, Th., Kerr, J. A., Pilling, M. J., Stocker, D., Troe, J., Tsang, W., Walker, R.W., and Warnatz, J., J. Phys. Chem. Ref. Data, 34, 757 (2005); “Supplement I,” Baulch, D. L., Cobos, C., Cox, R. A., Frank, P., Hayman, G., Just, Th., Kerr, J. A., Murrells, T., Murrells, M. J., Pilling, M. J., Troe, J., Walker, R.W., and Warnatz, J., J. Phys. Chem. Ref. Data, 23, 847(1994); “Evaluated Kinetic Data for Combustion Modeling,” Baulch, D. L., Cobos, C. J., Cox, R. A., Esser, C., Frank, P., Just, Th., Kerr, J. A., Pilling, M. J., Troe, J., Walker, R. W., and Warnatz, J., J. Phys. Chem. Ref. Data, 21, 411(1992). 2. Gas-Phase Combustion Chemistry, Gardiner, W. C., Jr., Springer-Verlag, NY, 1999: “Chapter 2. Combustion Chemistry of Nitrogen,” Dean, A. M.,

Appendix C: Specific Reaction Rate Constants

3.

4.

5.

6.

7.

691

and Bozzelli, J. W.; “Chapter 3. Kinetics and Mechanism of the Oxidation of Gaseous Sulfur Compounds,” Hynes, A. J., and Wine, P. H.; “Survey of Rate Coefficients in the C¶H¶Cl¶O System,” Senkan, S. M.; “Chapter 5. Ideal Gas Thermochemical Data for Combustion and Air Pollution Use,” Burcat, A., and Gardiner, W. C. Jr. Chemical Kinetic Data Base for Propellant Combustion: “I. Reactions Involving NO, NO2, HNO, HNO2, HCN, and N2O,” Tsang, W., and Herron, J. T., J. Phys. Chem. Ref. Data, 20, 609 (1991); “II. Reactions Involving CN, NCO, and HNCO,” Tsang, W., J. Phys. Chem. Ref. Data, 21, 750 (1992). Chemical Kinetic Data for Combustion Chemistry: “Part 1. Methane and Related Compounds,” Tsang, W., and Hampson, R. F., J. Phys. Chem. Ref. Data, 15, 1087 (1986); “Part 2. Methanol,” Tsang, W., J. Phys. Chem. Ref. Data, 16, 471 (1987); “Part 3. Propane,” Tsang, W., J. Phys. Chem. Ref. Data, 17, 887 (1988); “Part 4. Isobutane,” Tsang, W., J. Phys. Chem. Ref. Data, 19, 1 (1990); “Part 5. Propene,” J. Phys. Chem. Ref. Data, 20, 221 (1991). Combustion Chemistry, Gardiner, W. C., Jr., ed., Springer-Verlag, NY, 1985: “Chapter 5. Rate Coefficients in the C/H/O System,” Warnatz, J.; “Chapter 6. Survey of Rate Constants in the N/H/O System,” Hanson, R. K., and Salimian, S. Chemical Kinetic Data Sheets for High Temperature Reactions: “Part I,” N. Cohen and K.R. Westberg, J. Phys. Chem. Ref. Data, 12, 531 (1983); “Part II,” J. Phys. Chem. Ref. Data, 20, 1211 (1991). Evaluated Kinetic and Photochemical Data for Atmospheric Chemistry: “Supplement VI.” IUPAC Subcommittee on Gas Kinetic Data Evaluation for Atmospheric Chemistry, Atkinson, R., Baulch, D. L., Cox, R. A., Hampson, R. F., Jr., Kerr, J. A., Rossi, M. J., and Troe, J., J. Phys. Chem. Ref. Data, 26, 1329 (1997); “Supplement IV.” IUPAC Subcommittee on Gas Kinetic Data Evaluation for Atmospheric Chemistry, Atkinson, R., Baulch, D. L., Cox, R. A., Hampson, R. F., Jr., Kerr, J. A., and Troe, J., J. Phys. Chem. Ref. Data, 21, 1125 (1992); “Supplement III.” J. Phys. Chem. Ref. Data, 18, 881 (1989); “Supplement II.” J. Phys. Chem. Ref. Data, 13, 1259 (1984); “Supplement I.” J. Phys. Chem. Ref. Data, 11, 327 (1982); J. Phys. Chem. Ref. Data, 9, 295 (1980).

In order to take advantage of the developing cyber infrastructure, the Process Information Model (PrIMe) has recently been introduced for developing predictive models of chemical reaction systems based on the scientific collaboratory paradigm. PrIMe makes use of advances in computer science that allow assembly and manipulation of large amounts of combustion chemistry data that may be distributed over different sources using web-based computer networks. PrIMe consists of a data depository, a data library, and a set of computer-based tools for processing and assembling the data. More information can be found at http://primekinetics.org/.

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Appendix D

Bond Dissociation Energies of Hydrocarbons The bond dissociation energies that follow are taken from the review of McMillan and Golden [Ann. Rev. Phys. Chem. 33, 493 (1982)]. The reader should refer to this publication for the methods of determining the values presented, their uncertainty, and the original sources. In the tables presented, all bond energies and heats of formation are in kJ/mol. The values listed in the first column are the heats of formation at 298 K for the reference radical and those above the column heading for the associated radical. Thus, the tables presented are not only a source of bond energies, but also of heats of formation of radicals. McMillan and Golden employ the commonly invoked uncertainty of 5 kJ/mol (1kcal/mol) for dissociation energies and most of the uncertainties for the heats of formation fall in the same range. The reader is urged to refer to the McMillan and Golden reference for the specific uncertainty values.

693

694

TABLE D1 Bond Dissociation Energies of Alkanes* ΔH°f (R)

R

147

CH3

440

378

359

359

352

426

317

108

C2H5

411

359

344

339

331

408

300

(218) H

CH3

C2H5

i-C3H7

t-C4H9

(329) C6H5

(200) PhCH2

88

n-C3H7

410

362

344

336

330

409

302

76

i-C3H7

398

359

339

331

316

401

298

54

s-C4H9

400

356

335

328

–

–

–

36

t-C4H9

390

352

331

316

298

–

291

36

CH2C(CH3)3

418

346

346

–

–

–

–

280

Cyclopropyl

445

–

–

–

–

–

–

214

Cyclopropyl-methyl

408

–

–

–

–

–

–

214

Cyclobutyl

404

–

–

–

–

–

–

102

Cyclopentyl

395

–

–

–

–

–

–

58

Cyclohexyl

400

–

–

–

–

–

–

51

Cycloheptyl

387

–

–

–

–

–

–

*Note that in alkanes, values of 410, 397, and 389 kJ/mol characterize primary, secondary, and tertiary C¶H bonds, respectively.

Combustion

695

Appendix D: Bond Dissociation Energies of Hydrocarbons

TABLE D2 Bond Dissociation Energies of Alkenes, Alkynes, and Aromatics ΔH°f (R) 295

R C

C

329

565 548 164

C

C

• C6F5

C

C

126

C C

127

C

C

C

460

421

464

426

552

526

477

–

361

311

358

305

345

–

323

285

319

–

326

–

371

–

344

–

C

C C

CH3

C

C C

77

C

H

C

C 38

C

C

C

C

C

C 40

C

C

C

C

C

C –

Cl C

161

C

C

696

Combustion

TABLE D2 (continued) ΔH°f (R)

R

205

C

C

C

C

H

CH3

318

–

C

205

C

C

C

C

–

C

242

297

–

1977

305

–

271

305

–

440

379

–

368

317

356

305

200

C

253

C

338

C

342

283

311

C

356

305

357

312

353

308

169

139

C

C

C

C

C

(continues)

697

Appendix D: Bond Dissociation Energies of Hydrocarbons

TABLE D2 (continued) ΔH°f (R)

R

–

O

H

CH3

362

314

351

–

374

318

365

308

365

321

344

303

339

296

348

305

C

–

341

C

C

C

294

C

C

C

273

C C

222

C

C

C

C

C C

C

C

C

C 257

C C

C

C

C

C

C 295

C C

698

Combustion

TABLE D3 Bond Dissociation Energies of C/H/O Compounds Oxygen-centered radicals ΔH°f (R)

R1

(218) H

R1

(39) OH

(147) CH3

(108) C2H5

39

OH

498

213

213

386

383

18

OCH3

437

157

–

349

342

17

OC2H5

436

159

–

346

344

41

O-n-C3H7

433

155

–

343

–

63

O-n-C4H9

431

–

–

–

–

52

O-i-C3H7

438

158

—

346

–

69

O-s-C4H9

441

152

–

–

–

91

O-t-C4H9

440

159

–

348

–

–

O-t-C5H11

–

164

–

–

–

–

OCH2C(CH3)3

428

152

194

–

–

48

OC6H5

362

–

–

267

264

–

OCF3

–

193

–

–

–

–

OC(CF3)3

–

149

–

–

–

10

O2H

365

–

–

–

–

208

O2CCH3

443

127

–

–

346

228

O2CC2H5

445

127

–

–

–

249

O2C-nC3H7

443

127

–

–

–

(329) C6H5

R1

Carbon-centered radicals ΔH°f (R) 37

R1

(218) H

(147) CH3

CHO

364

345

403

286

COCH3

360

340

391

282

COCH ¨ CH2

364

–

–

–

43

COC2H5

366

337

395

–

109

COC6H5

364

343

377

278

–

COCF3

381

–

–

–

24

CH2COCH3

411

361

–

–

70

CH(CH3)COCH3

386

–

–

–

24 72

(continues)

699

Appendix D: Bond Dissociation Energies of Hydrocarbons

TABLE D3 (continued) Carbon-centered radicals (218) H

(329) C6H5

R1

–

403

336

389

–

–

355

C(OH)(CH3)2

381

–

–

–

12

CH2OCH3

389

361

–

–

18

Tetrahydrofuran–2–yl

385

–

–

–

CH(OH)CH¨CH2

341

–

–

–

169

COOCH3

388

–

–

–

70

CH2OCOC6H5

419

–

–

–

223

COOHßCH2C6H5

280

–

–

–

248

(C6H5)2CHßCOOH

249

–

–

–

–

C6H5CH2COßCH2C6H5

274

–

–

–

200

C6H5CH2ßOH

340

–

–

–

48

C6H5COßF3

309

–

–

–

ΔH°f (R)

R1

26

CH2OH

393

64

CH(OH)CH3

111

0.0

(147) CH3

TABLE D4 Bond Dissociation Energies of SulfurContaining Compounds R1–R2

0 D298

ΔH°f (R)

HSßH

381

141

CH3SßH

379

139

RSßH

381

–

CH3ßSH

310

–

C2H5ßSH

295

–

t-BußSH

286

–

C6H5ßSH

362

–

TABLE D4 Bond Dissociation Energies of SulfurContaining Compounds 0 D298

ΔH°f (R)

CH3SßCH3

323

–

CH3SßC2H5

307

–

CH3Sßn-C3H7

309

–

PhSßH

349

–

PhSßCH3

290

230

PhCH2ßSCH3

257

–

CSßS

431

272

OSßO

544

–

CH3SO2ßCH3

279

–

CH3SO2ßCH2CH ¨ CH3

208

–

CH3SO2ßCH2C6H5

221

–

RS2ßH

293

–

RS2ßCH3

238

–

HSßSH

276

–

RSßSR

301

–

R1–R2

TABLE D5 Bond Dissociation Energies of Nitrogen-Containing Compounds Amines and nitriles (218) H

(147) CH3

(108) C2H5

(200) PhCH2

(329) C6H5

(189) NH2

NH2

449

355

341

297

427

275

177

NHCH3

418

344

334

287

421

268

145

N(CH3)2

383

316

303

260

390

247

237

NHC6H5

368

299

289

–

339

219

233

N(CH3)C6H5

366

296

–

–

–

–

ΔH°f (R)

R

185

(continues)

701

Appendix D: Bond Dissociation Energies of Hydrocarbons

TABLE D5 (continued) Amines and nitriles ΔH°f (R)

R

(218) H

(147) CH3

(108) C2H5

(200) PhCH2

(329) C6H5

(189) NH2

33

NF2

317

–

–

–

–

–

469

N3

385

–

–

–

–

–

149

CH2NH2

390

344

332

285

390

–

126

CH2NH(CH3)

364

320

–

–

367

–

109

CH2N(CH3)2

351

308

–

–

352

–

435

CN

518

510

495

–

548

–

245

CH2CN

389

340

322

–

–

–

209

CH(CH3)CN

376

330

–

–

–

–

167

C(CH3)2CN

362

313

–

–

–

–

249

C(CH3)(CN)C6H5

–

251

–

–

–

–

Nitro and nitroso compounds and nitrates ΔH°f (R)

R

(90) NO

(33) NO2

(71) ONO2

218

H

–

328

423

39

OH

206

207

163

147

CH3

167

254

–

108

i-C3H5

–

245

–

76

i-C3H7

153

247

–

36

t-C4H9

165

245

–

CF3

179

–

–

79

CCl3

134

–

–

331

C6H5

213

298

–

548

C6F5

208

–

–

467

–

C(NO2)R2

–

204

–

–

C(NO2)2R

–

183

–

–

C(NO2)3

–

169

–

–

RO

171

170

–

33

NO2

41

57

–

702

Combustion

TABLE D6 Bond Dissociation Energies of Halocarbons ΔH°f (R)

R

147

(218) H

(147) CH3

(79) F

(121) Cl

(112) Br

(107) I

(465) CF3

CH3

440

378

460

354

297

239

425

108

C2H5

411

–

451

334

284

223

–

76

i-C3H7

398

359

446

338

286

224

–

200

CH2C6H5

368

317

–

302

241

202

–

33

CH2F

418

498

357

357

–

–

396

248

CHF2

423

400

527

–

289

–

–

467

CF3

446

425

546

361

295

230

413

118

CH2Cl

422

–

–

335

–

–

–

101

CHCl2

414

–

–

325

–

–

–

79

CCl3

401

–

426

306

231

–

–

269

CF2Cl

425

–

515

318

270

–

–

96

CFCl2

–

–

460

305

–

–

–

174

CH2Br

427

–

–

–

–

–

–

227

CHBr2

434

–

–

–

–

–

–

–

CBr3

402

–

–

–

235

–

–

893

C2F5

430

–

531

346

287

214

–

–

n-C3F7

435

–

–

–

278

208

–

–

i-C3F7

431

–

–

–

274

–

–

303

CF2CH3

416

–

522

–

–

218

–

517

CH2CF3

446

–

458

–

–

236

–

–

CHClCF3

426

–

–

–

275

–

–

–

CClBrCF3

404

–

–

–

251

–

–

435

CN

518

510

470

422

367

303

561

39

OH

498

387

–

251

234

234

–

331

C6H5

466

428

526

400

337

274

–

548

C6F5

487

–

–

383

–

277

–

Appendix E

Flammability Limits in Air The data presented in Table E1 are for fuel gases and vapors and are taken almost exclusively from Zabetakis. [US Bur. Mines Bull. 627 (1965)]. The conditions are for the fuel–air mixture at 25°C and 1 atm unless otherwise specified. As noted in the text, most fuels have a rich limit at approximately φ  3.3 and a lean limit at approximately φ  0.5. The fuels which vary most from the rich limit are those that are either very tightly bound as ammonia is or which can decompose as hydrazine or any monopropellant does. Additional sources of flammability limit data can be found in the Ignition Handbook by Babrauskas (Fire Science Publishers, Issaquah, WA, 2003) and from Kuchta [US Bur. Mines Bull. 680 (1985)]. There can also be a flammability limit associated with dust clouds. The flammability limits of combustible dusts are reported as the minimum explosion concentrations. The upper explosion limits for dust clouds have not been determined due to experimental difficulties. In the 14th edition of the Fire Protection Handbook (National Fire Protection Association, Boston, MA, 1975), numerous results from the US Bureau of Mines reports are listed. These results were obtained with dusts 74 μm or smaller. It should be noted that variations in minimum explosive concentrations will occur with change in particle diameter, that is, the minimum explosive concentration is lowered as the diameter of the particle decreases. Other conditions which affect this limit are sample purity, oxygen concentration, strength of ignition source, turbulence, and uniformity of the dispersion. The NFPA tabulation is most extensive and includes data for dusts from agricultural materials, carbonaceous matter, chemicals, drugs, dyes, metals, pesticides, and various plastic resins and molding compounds. Except for metal dusts, it is rather remarkable that most materials have a minimum explosive concentration in the range 0.03–0.05 kg/m3. It should be noted, however, that the variation according to the specific compound can range from 0.01 to 0.50 kg/m3. For a specific value the reader should refer to the NFPA handbook.

703

704

Combustion

TABLE E1 Flammability Limits of Fuel Gases and Vapors in Air at 25°C and 1 atm Lean limit

Rich limit (Vol.%)/ (Vol.%)ST, [φ]

Vol.%

(Vol.%)/ (Vol.%)ST, [φ]

Fuel

Vol.%

Acetal

1.6

Acetaldehyde

4.0

Acetic acid

5.4 (100°C)

Acetic anhydride

2.7 (47°C)

Acetanilide

1.0 (calc)

Acetone

2.6

Acetophenone

1.1 (calc)

Acetylacetone

1.7 (calc)

Acetyl chloride

5.0 (calc)

Acetylene

2.5

100

Acrolein

2.8

31

Acrylonitrile

3.0

Acetone cyanohydrin

2.2

Adipic acid

1.6 (calc)

Aldol

2.0 (calc)

Allyl alcohol

2.5

18

Allyl amine

2.2

22

Allyl bromide

2.7 (calc)

Allyl chloride

2.9

o-Aminodiphenyl

0.66

Ammonia

15

0.69 [0.63]

28

1.3 [1.4]

n-Amyl acetate

1.0 (100°C)

0.51

7.1 (100°C)

3.3

n-Amyl alcohol

1.4 (100°C)

0.51

10 (100°C)

t-Amyl alcohol

1.4 (calc)

n-Amyl chloride

1.6 (50°C)

t-Amyl chloride

1.5 (85°C)

10 0.52

6

0

10 (75°C)

0.52

13

2.6

12

4.1

8.6 (100°C)

705

Appendix E: Flammability Limits in Air

TABLE E1 (continued) Lean limit

Rich limit (Vol.%)/ (Vol.%)ST, [φ]

Fuel

Vol.%

n-Amyl ether

0.7 (calc)

Amyl nitrite

1.0 (calc)

n-Amyl propionate

1.0 (calc)

Amylene

1.4

8.7

Aniline

1.2 (140°C)

8.3 (140°C)

Anthracene

0.65 (calc)

n-Amyl nitrate

1.1

Benzene

1.3 (100°C)

Benzyl benzoate

0.7 (calc)

Benzyl chloride

1.2 (calc)

Bicyclohexyl

0.65 (100°C)

Biphenyl

0.70 (110°C)

2-Biphenyl amine

0.8 (calc)

Bromobenzene

1.6 (calc)

Butadiene (1,3)

2.0

0.54

12

3.3

n-Butane

1.8

0.58 [0.57]

8.4

2.7 [2.8]

1,3-Butandiol

1.9 (calc)

Butene-1

1.6

0.50

10

2.9

Butene-2

1.7

0.53

9.7

2.9

n-Butyl acetate

1.4 (50°C)

0.55

8.0 (100°C)

3.1

n-Butyl alcohol

1.7 (100°C)

0.5

12 (100°C)

s-Butyl alcohol

1.7 (100°C)

9.8 (100°C)

t-Butyl alcohol

1.9 (100°C)

9.0 (100°C)

t-Butyl amine

1.7 (100°C)

8.9 (100°C)

n-Butyl benzene

0.82 (100°C)

5.8 (100°C)

s-Butyl benzene

0.77 (100°C)

5.8 (100°C)

t-Butyl benzene

0.77 (100°C)

5.8 (100°C)

0.48

Vol.%

(Vol.%)/ (Vol.%)ST, [φ]

7.9 (100°C)

2.9

5.1 (150°C)

(continues)

706

Combustion

TABLE E1 (continued) Lean limit

Rich limit (Vol.%)/ (Vol.%)ST, [φ]

Fuel

Vol.%

Vol.%

n-Butyl bromide

2.5 (100°C)

Butyl cellosolve

1.1 (150°C)

11 (175°C)

n-Butyl chloride

1.8

10 (100°C)

n-Butyl formate

1.7

n-Butyl stearate

0.3 (calc)

Butyric acid

2.1 (calc)

γ-Butyrolactone

2.0 (150°C)

Carbon disulfide

1.3

Carbon monoxide

12.5

Chlorobenzene

1.4

m-Cresol

1.1 (150°C)

Crotonaldehyde

2.1

Cumene

0.88 (100°C)

Cyanogen

6.6

Cyclobutane

1.8 (calc)

0.56

Cycloheptane

1.1 (calc)

Cyclohexane

1.3

Cyclohexanol

1.2 (calc)

Cyclohexene

1.2 (100°C)

Cyclohexyl acetate

1.0 (calc)

Cyclopentane

1.5 (calc)

0.55

Cyclopropane

2.4

Cymene

(Vol.%)/ (Vol.%)ST, [φ]

0.54

8.2

2.6

0.2

50

7.7

74

16 (60°C) 0.51

6.5 (100°C)

3.8

0.56

6.7 (calc)

3.4

0.57

7.8

3.4

0.54

10.4

2.3

0.85 (100°C)

0.56

6.5 (100°C)

3.6

Decaborane

0.2 (calc)

0.11

Decalin

0.74 (100°C)

n-Decane

0.75 (53°C)

Deuterium

4.9

Diborane

0.8

4.9 (100°C) 0.56

5.6 (86°C)

4.2

75 0.12

88

13.5

707

Appendix E: Flammability Limits in Air

TABLE E1 (continued) Lean limit

Rich limit (Vol.%)/ (Vol.%)ST, [φ]

Fuel

Vol.%

Diethyl amine

1.8

Diethyl aniline

0.8 (calc)

1,4-Diethyl benzene

0.8 (100°C)

Diethyl cyclohexane

0.75

Diethyl ether

1.9

3,3-Diethyl pentane

0.7 (100°C)

Diethyl ketone

1.6 (calc)

Diisobutyl carbinol

0.82 (100°C)

6.1 (175°C)

Diisobutyl ketone

0.79 (100°C)

6.2 (100°C)

Diisopropyl ether

1.4

Dimethyl amine

2.8

2,2-Dimethyl butane

1.2

7.0

2,3-Dimethyl butane

1.2

7.0

Dimethyl decalin

0.69 (100°C)

5.3 (110°C)

Dimethyl dichlorosilane

3.4

Dimethyl ether

3.4

N, N-Dimethyl formamide

1.8 (100°C)

14 (100°C)

2,3-Dimethyl pentane

1.1

6.8

2,2-Dimethyl propane

1.4

7.5

Dimethyl sulfide

2.2

Dioxane

2.0

22

Dipentene

0.75 (150°C)

6.1 (150°C)

Diphenyl amine

0.7 (calc)

Diphenyl ether

Vol.%

(Vol.%)/ (Vol.%)ST, [φ]

10

0.56

36

11

0.55

0.57

0.52

0.50

7.9

27

20

3.5

4.1

4.5

0.8 (calc)

Diphenyl methane

0.7 (calc)

Divinyl ether

1.7

0.42

27

6.7 (continues)

708

Combustion

TABLE E1 (continued) Lean limit

Rich limit

Fuel

Vol.%

(Vol.%)/ (Vol.%)ST, [φ]

Vol.%

(Vol.%)/ (Vol.%)ST, [φ]

n-Dodecane

0.60 (calc)

0.54

Ethane

3.0

0.53 [0.52]

12.4

2.2 [2.4]

Ethyl acetate

2.2

0.55

11

2.7

Ethyl alcohol

3.3

0.5

19 (60°C)

2.9

Ethyl amine

3.5

Ethyl benzene

1.0 (100°C)

0.51

6.7 (100°C)

3.4

Ethyl chloride

3.8

Ethyl cyclobutane

1.2

0.53

7.7

3.4

Ethyl cyclohexane

0.95 (130°C)

0.56

6.6 (130°C)

3.9

Ethyl cyclopentane

1.1

0.56

6.7(0.5 atm)

3.4

Ethyl formate

2.8

0.50

16

2.8

Ethyl lactate

1.5

Ethyl mercaptan

2.8

0.63

18

4.4

Ethyl nitrate

4.0

Ethyl nitrite

3.0

Ethyl propionate

1.8

0.58

11

3.5

Ethyl propyl ether

1.7

0.62

9

3.3

Ethylene

2.7

0.41 [0.40]

36

5.5 [8.0]

Ethyleneimine

3.6

Ethylene glycol

3.5 (calc)

Ethylene oxide

3.6

100

Furfural alcohol

1.8 (72°C)

16 (117°C)

Gasoline (100/130)

1.3

7.1

Gasoline (115/145)

1.2

7.1

n-Heptane

1.05

0.56

n-Hexadecane

0.43 (calc)

0.51

n-Hexane

1.2

0.56

n-Hexyl alcohol

1.2 (100°C)

0.53

50

46

6.7

3.6

7.4

3.4

709

Appendix E: Flammability Limits in Air

TABLE E1 (continued) Lean limit

Rich limit (Vol.%)/ (Vol.%)ST, [φ]

Vol.%

(Vol.%)/ (Vol.%)ST, [φ]

Fuel

Vol.%

n-Hexyl ether

0.6 (calc)

Hydrazine

4.7

0.27

100

5.8

Hydrogen

4.0

0.14 [0.10]

75

2.54 [7.14]

Hydrogen cyanide

5.6

Hydrogen sulfide

4.0

Isoamyl acetate

1.1 (100°C)

7.0 (100°C)

Isoamyl alcohol

1.4 (100°C)

9.0 (100°C)

Isobutane

1.8

8.4

Isobutyl alcohol

1.7 (100°C)

11 (100°C)

Isobutyl benzene

0.82 (100°C)

6.0 (175°C)

Isobutyl formate

2.0

8.9

Isobutylene

1.8

Isopentane

1.4

Isophorone

0.84

Isopropylacetate

1.7 (calc)

Isopropyl alcohol

2.2

Isopropyl biphenyl

0.6 (calc)

Jet fuel (JP-4)

1.3

Methane

5.0

0.53 [0.50]

15.0

1.6 [1.7]

Methyl acetate

3.2

0.57

16

2.8

Methyl acetylene

1.7

Methyl alcohol

6.7

0.55 [0.51]

36 (60°C)

2.9 [4.0]

Methylamine

4.2 (calc)

Methyl bromide

10

3-Methyl-1-butene

1.5

0.55

9.1

3.3

Methyl butyl ketone

1.2 (50°C)

0.58

8.0 (100°C)

3.3

Methyl cellosolve

2.5 (125°C)

40 0.33

0.53

44

9.6

3.6

2.8

8

15

20 (140°C) (continues)

710

Combustion

TABLE E1 (continued) Lean limit

Rich limit (Vol.%)/ (Vol.%)ST, [φ]

Vol.%

(Vol.%)/ (Vol.%)ST, [φ]

0.56

6.7 (calc)

3.4

Fuel

Vol.%

Methyl cellosolve acetate

1.7 (150°C)

Methyl ethyl ether

2.2 (calc)

Methyl chloride

7 (calc)

Methyl cyclohexane

1.1

Methyl cyclopentadiene

1.3 (100°C)

Methyl ethyl ketone

1.9

0.52

10

2.7

Methyl formate

5.0

0.53

23

2.4

Methyl cycolhexanol

1.0 (calc)

Methyl isobutyl carbinol

1.2 (calc)

Methyl isopropenyl ketone

1.8 (50°C)

Methyl lactate

2.2 (100°C)

Methyl mercaptan

3.9

1-Methyl naphthalene

0.8 (calc)

2-Methyl pentane

1.2 (calc)

Methyl propionate

7.6 (100°C)

9.0 (50°C)

0.60

22

3.4

2.4

0.60

13

3.2

Methyl propyl ketone

1.6

0.55

8.2

2.8

Methyl styrene

1.0 (calc)

Methyl vinyl ether

2.6

39

Monoisopropyl bicyclohexyl

0.52

4.1 (200°C)

2-Monoisopropyl biphenyl

0.53 (175°C)

3.2 (200°C)

Monomethylhydrazine

4

Naphthalene

0.88 (78°C)

Nicotine

0.75 (calc)

Nitroethane

3.4

Nitromethane

7.3

0.52 5.9 (122°C)

711

Appendix E: Flammability Limits in Air

TABLE E1 (continued) Lean limit

Rich limit (Vol.%)/ (Vol.%)ST, [φ]

Fuel

Vol.%

1-Nitropropane

2.2

2-Nitropropane

2.5

n-Nonane

0.85 (43°C)

0.58

n-Octane

0.95

0.58

Paraldehyde

1.3

0.48

Pentaborane

0.42

0.12

n-Pentadecane

0.50 (calc)

0.52

n-Pentane

1.4

0.55

Phthalic anhydride

1.2 (140°C)

3-Picoline

1.4 (calc)

Pinane

0.74 (160°C)

Propadiene

2.16

Propane

2.1

1,2-Propanediol

2.5 (calc)

β-Propiolactone

2.9 (75°C)

Propionaldehyde

Vol.%

(Vol.%)/ (Vol.%)ST, [φ]

7.8

3.1

9.2 (195°C)

7.2 (160°C)

0.57 [0.56]

9.5

2.5 [2.7]

2.9

0.59

17

n-Propyl acetate

1.8

0.58

8 (90°C)

2.6

n-Propyl alcohol

2.2 (53°C)

0.49

14 (100°C)

3.2

Propylamine

2.0

Propyl chloride

2.4 (calc)

n-Propyl nitrate

1.8 (125°C)

Propylene

2.4

Propylene dichloride

3.1 (calc)

Propylene glycol

2.6 (96°C)

Propylene oxide

2.8

37

Pyridine

1.8 (60°C)

12 (70°C)

Propargyl alcohol

2.4 (50°C)

100 (125°C) 0.54 [0.53]

11

2.5 [2.7]

(continues)

712

Combustion

TABLE E1 (continued) Lean limit

Rich limit (Vol.%)/ (Vol.%)ST, [φ]

Vol.%

(Vol.%)/ (Vol.%)ST, [φ]

Fuel

Vol.%

Quinoline

1.0 (calc)

Styrene

1.1 (29°C)

Sulfur

2.0 (247°C)

p-Terphenyl

0.96 (calc)

Tetraborane

0.4 (calc)

0.11

n-Tetradecane

0.5 (calc)

0.52

Tetrahydrofurane

2.0

Tetralin

0.84 (100°C)

2,2,3,3-Tetramethyl pentane

0.8

Toluene

1.2 (100°C)

Trichloroethylene

12 (30°C)

n-Tridecane

0.55 (calc)

Triethylamine

1.2

8.0

Triethylene glycol

0.9 (150°C)

9.2 (203°C)

2,2,3-Trimethyl butane

1.0

Trimethyl amine

2.0

2,2,4-Trimethyl pentane

0.95

Trimethylene glycol

1.7 (calc)

Trioxane

3.2 (calc)

Turpentine

0.7 (100°C)

n-Undecane

0.68 (calc)

0.56

UDMH

2.0

0.40

Vinyl acetate

2.6

Vinyl chloride

3.6

m-Xylene

1.1 (100°C)

0.56

6.4 (100°C)

3.3

o-Xylene

1.1 (100°C)

0.56

6.4 (100°C)

3.3

p-Xylene

1.1 (100°C)

0.56

6.6 (100°C)

3.4

5.0 (150°C)

0.53

7.1 (100°C)

3.1

40 (70°C) 0.53

12

95

19.1

33

Appendix F

Laminar Flame Speeds The compilation of laminar flame speed data given in Tables F1 and F2 is due to Gibbs and Calcote [J. Chem. Eng. Data 4, 226 (1959)]. The reader is referred to the quoted paper for details on the chosen values. The data are for premixed fuel–air mixtures at 25°C and 100°C and 1 atm pressure. Examples of more recent data (obtained from a counter-flow, double-flame, burner configuration vs. the Bunsen cone burner configuration) have also been included from Law (in Reduced Kinetic Mechanisms for Applications in Combustion Systems, N. Peters and B. Rogg, eds., Springer-Verlag, NY, 1993) and Vagelopoulos, Egolfopoulous and Law [Proceedings of the Combustion Institute 25, 1341–1347 (1994)]. The values of Law and co-workers are denoted by the letters (L) and (L2) following the flame speed. Table F3 from Law reports flame speed data as a function of pressure.

713

φ  0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

Smax

φ at Smax

30.6

36.0

40.6

44.5

47.3

47.3

44.4

37.4

47.6

1.14

22.0 (L)

29.0 (L)

36.5 (L)

42.5 (L)

43.0 (L)

42.5 (L)

40.0 (L)

27.5 (L)

–

–

–

–

42.3

45.6

46.2

42.4

34.3

–

46.4

1.06

24.0 (L)

32.0 (L)

39.5 (L)

44.0 (L)

45.0 (L)

43.5 (L)

37.0 (L)

28.0 (L)

–

–

23.0 (L2)

30.0 (L2)

37.0 (L2)

39.0 (L2)

41.0 (L2)

40.5 (L2)

33.5 (L2)

25.0 (L2)

–

–

n-Butane

–

38.0

42.6

44.8

44.2

41.2

34.4

25.0

44.9

1.03

Methane

–

30.0

38.3

43.4

44.7

39.8

31.2

–

44.8

1.08

20.5 (L)

28.0 (L)

36.0 (L)

40.5 (L)

42.0 (L)

37.0 (L)

27.0 (L)

17.5 (L)

–

–

17 (L2)

25.0 (L2)

33.0 (L2)

38 (L2)

38.5 (L2)

34.0 (L2)

24.0 (L2)

13.5 (L2)

–

–

n-Pentane

–

35.0

40.5

42.7

42.7

39.3

33.9

–

43.0

1.05

n-Heptane

–

37.0

39.8

42.2

42.0

35.5

29.4

–

42.8

1.05

2,2,4-Trimethylpentane

–

37.5

40.2

41.0

37.2

31.0

23.5

–

41.0

0.98

2,2,3-Trimethylpentane

–

37.8

39.5

40.1

39.5

36.2

–

–

40.1

1.00

2,2-Dimethylbutane

–

33.5

38.3

39.9

37.0

33.5

–

–

40.0

0.98

Isopentane

–

33.0

37.6

39.8

38.4

33.4

24.8

–

39.9

1.01

2,2-Dimethylpropane

–

–

31.0

34.8

36.0

35.2

33.5

31.2

36.0

1.10

Fuel

714

TABLE F1 Burning Velocities of Various Fuels at 25°C Air-Fuel Temperature (0.31 mol.% H2O in Air). Burning Velocity S as a Function of Equivalence Ratio φ in cm/s

Saturated hydrocarbons Ethane

Propane

Combustion

Acetylene

–

107

130

144

151

154

154

152

155

1.25

–

107 (L)

–

136 (L)

–

151 (L)

–

155 (L)

–

–

37.0

50.0

60.0

68.0

73.0

72.0

66.5

60.0

73.5

1.13

37.0 (L)

48.0 (L)

60.0 (L)

66.0 (L)

70.0 (L)

72.0 (L)

71.0 (L)

65.0 (L)

–

–

Propylene

–

62.0

66.6

70.2

72.2

71.2

61.0

–

72.5

1.14

1,3-Butadiene

–

–

42.6

49.6

55.0

57.0

56.9

55.4

57.2

1.23

n-1-Heptene

–

46.8

50.7

52.3

50.9

47.4

41.6

–

52.3

1.00

Propylene

–

–

48.4

51.2

49.9

46.4

40.8

–

51.2

1.00

n-2-Pentene

–

35.1

42.6

47.8

46.9

42.6

34.9

–

48.0

1.03

2,2,4-Trimethyl-3-pentene

–

34.6

41.3

42.2

37.4

33.0

–

–

42.5

0.98

–

34.5

42.0

48.0

50.2

47.5

44.4

42.2

50.4

1.08

21.5 (L)

31.0 (L)

37.5 (L)

48.0 (L)

54.0 (L)

53.5 (L)

48.0 (L)

42.0 (L)

@45°C

–

Isopropyl alcohol

–

34.4

39.2

41.3

40.6

38.2

36.0

34.2

41.4

1.04

Triethylamine

–

32.5

36.7

38.5

38.7

36.2

28.6

–

38.8

1.06

n-Butyl chloride

24.0

30.7

33.8

34.5

32.5

26.9

20.0

–

34.5

1.00

Ethylene

Appendix F: Laminar Flame Speeds

Unsaturated hydrocarbons

Substituted alkyls Methanol

(Continues)

715

716

TABLE F1 (continued) Fuel

φ  0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

Smax

φ at Smax

Allyl chloride

30.6

33.0

33.7

32.4

29.6

–

–

–

33.8

0.89

Isopropyl mercaptan

–

30.0

33.5

33.0

26.6

–

–

–

33.8

0.94

Ethylamine

–

28.7

31.4

32.4

31.8

29.4

25.3

–

32.4

1.00

Isopropylamine

–

27.0

29.5

30.6

29.8

27.7

–

–

30.6

1.01

n-Propyl chloride

–

24.7

28.3

27.5

24.1

–

–

–

28.5

0.93

Isopropyl chloride

–

24.8

27.0

27.4

25.3

–

–

–

27.6

0.97

n-Propyl bromide

No ignition

Silanes Tetramethylsilane

39.5

49.5

57.3

58.2

57.7

54.5

47.5

–

58.2

1.01

Trimethylethoxysilane

34.7

41.0

47.4

50.3

46.5

41.0

35.0

–

50.3

1.00

Acrolein

47.0

58.0

66.6

65.9

56.5

–

–

–

67.2

0.95

Propionaldehyde

–

37.5

44.3

49.0

49.5

46.0

41.6

37.2

50.0

1.06

Acetaldehyde

–

26.6

35.0

41.4

41.4

36.0

30.0

–

42.2

1.05

–

40.4

44.2

42.6

38.2

–

–

–

44.4

0.93

Aldehydes

Acetone

Combustion

Ketones

–

36.0

42.0

43.3

41.5

37.7

33.2

–

43.4

0.99

Vinyl acetate

29.0

36.6

39.8

41.4

42.1

41.6

35.2

–

42.2

1.13

Ethyl acetate

–

30.7

35.2

37.0

35.6

30.0

–

–

37.0

1.00

Dimethyl ether

–

44.8

47.6

48.4

47.5

45.4

42.6

–

48.6

0.99

Diethyl ether

30.6

37.0

43.4

48.0

47.6

40.4

32.0

–

48.2

1.05

Dimethoxymethane

32.5

38.2

43.2

46.6

48.0

46.6

43.3

–

48.0

1.10

Diisopropyl ether

–

30.7

35.5

38.3

38.6

36.0

31.2

–

38.9

1.06

–

29.9

31.9

33.0

30.1

24.8

–

–

33.0

1.00

–

41.0

46.8

50.0

49.6

46.5

42.0

35.5

50.4

1.04

Furan

48.0

55.0

60.0

62.5

62.4

60.0

–

–

62.9

1.05

Benzene

–

39.4

45.6

47.6

44.8

40.2

35.6

–

47.6

1.00

Thiophene

33.8

37.4

40.6

43.0

42.2

37.2

24.6

–

43.2

1.03

57.2

70.7

83.0

88.8

89.5

87.2

81.0

73.0

89.5

1.07

Esters

Ethers

Appendix F: Laminar Flame Speeds

Methyl ethyl ketone

Thio ethers Dimethyl sulfide Peroxides Di-tert-butyl peroxide Aromatic Compounds

Cyclic compounds

(Continues)

717

Ethylene oxide

718

TABLE F1 (continued) Fuel

φ  0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

Smax

φ at Smax

Butadiene monoxide

–

6.6

47.4

57.8

64.0

66.9

66.8

64.5

67.1

1.24

Propylene oxide

41.6

53.3

62.6

66.5

66.4

62.5

53.8

–

67.0

1.05

Dihydropyran

39.0

45.7

51.0

54.5

55.6

52.6

44.3

32.0

55.7

1.08

Cyclopropane

–

40.6

49.0

54.2

55.6

53.5

44.0

–

55.6

1.10

Tetrahydropyran

44.8

51.0

53.6

51.5

42.3

–

–

–

53.7

0.93

Tetrahydrofuran

–

–

43.2

48.0

50.8

51.6

49.2

44.0

51.6

1.19

Cyclopendadiene

36.0

41.8

45.7

47.2

45.5

40.6

32.0

–

47.2

1.00

Ethylenimine

–

37.6

43.4

46.0

45.8

43.4

38.9

–

46.4

1.04

Cyclopentane

31.0

38.4

43.2

45.3

44.6

41.0

34.0

–

45.4

1.03

Cyclohexane

–

–

41.3

43.5

43.9

38.0

–

–

44.0

1.08

102

120

145

170

204

245

213

290

325

1.80

124 (L)

150 (L)

187 (L)

210 (L)

230 (L)

245 (L)

–

–

–

–

Carbon disulfide

50.6

58.0

59.4

58.8

57.0

55.0

52.8

51.6

59.4

0.91

Carbon monoxide

–

–

–

–

28.5

32.0

34.8

38.0

52.0

2.05

Hydrogen sulfide

34.8

39.2

40.9

39.1

32.3

–

–

–

40.9

0.90

Inorganic compounds Hydrogen

Combustion

Fuel

φ  0.7

0.8

0.9

1.0

Propargyl alcohol

–

76.8

100.0

110.0

110.5

108.8

Propylene oxide

74.0

86.2

93.0

96.6

97.8

94.0

Hydrazine*

87.3

90.5

93.2

94.3

93.0

90.7

Furfural

62.0

73.0

83.3

87.0

87.0

Ethyl nitrate

70.2

77.3

84.0

86.4

83.0

Butadiene monoxide

51.4

57.0

64.5

73.0

79.3

81.0

80.4

76.7

81.1

1.23

Carbon disulfide

64.0

72.5

76.8

78.4

75.5

71.0

66.0

62.2

78.4

1.00

n-Butyl ether

–

67.0

72.6

70.3

65.0

–

–

–

72.7

0.91

Methanol

50.0

58.5

66.9

71.2

72.0

66.4

58.0

48.8

72.2

1.08

31.5 (L)

43.0 (L)

59.5 (L)

63.5 (L)

66.0 (L)

65.0 (L)

61.5 (L)

51.0 (L)

@95°C

–

1.1

1.4

Smax

φ at Smax

105.0

85.0

110.5

1.08

84.0

71.5

97.9

1.09

87.4

83.7

94.4

0.98

84.0

77.0

65.5

87.3

1.05

72.3

–

–

86.4

1.00

1.2

1.3

Diethyl cellosolve

49.5

56.0

63.0

69.0

69.7

65.2

–

–

70.4

1.05

Cyclohexan monoxide

54.5

59.0

63.5

67.7

70.0

64.0

–

–

70.0

1.10

Epichlorohydrin

53.0

59.5

65.0

68.6

70.0

66.0

58.2

–

70.0

1.10

n-Pentane

–

50.0

55.0

61.0

62.0

57.0

49.3

42.4

62.9

1.05

n-Propyl alcohol

49.0

56.6

62.0

64.6

63.0

50.0

37.4

–

64.8

1.03

41.5

50.0

58.5

63.8

59.5

53.8

46.2

38.8

63.8

1.00

54.0

58.8

62.6

63.5

59.0

49.5

42.0

36.7

63.5

1.00

Pinene

48.5

58.3

62.5

62.1

56.6

50.0

–

–

63.0

0.95

Nitroethane

51.5

57.8

61.4

57.2

46.0

28.0

–

–

61.4

0.92

Iso-octane

–

50.2

56.8

57.8

53.3

50.5

–

–

58.2

0.98

Pyrrole

–

52.0

55.6

56.6

56.1

52.8

48.0

43.1

56.7

1.00

Aniline

–

41.5

45.4

46.6

42.9

37.7

32.0

–

46.8

0.98

Dimethyl formamide

–

40.0

43.6

45.8

45.5

40.7

36.7

–

46.1

1.04

*Results questionable because of an indication of decomposition in the stainless-steel feed system.

719

n-Heptane Ethyl nitrite

Appendix F: Laminar Flame Speeds

TABLE F2 Burning Velocities of Various Fuels at 100°C Air-Fuel Temperature (0.31 mol.% H2O in Air). Burning Velocity S as a Function of Equivalence Ratio φ in cm/s

720

Combustion

TABLE F3 Burning Velocities of Various Fuels in Air as a Function of Pressure for an Equivalence Ratio of 1 in cm/s Fuel

P  0.25 atm

0.5

1.0

2.0

3.0

Methane

59 (L)

50 (L)

40.5 (L)

29 (L)

22 (L)

Ethane

54 (L)

48 (L)

42.5 (L)

36.5 (L)

30 (L)

Propane

59 (L)

51 (L)

44.0 (L)

35.5 (L)

31 (L)

Appendix G

Spontaneous Ignition Temperature Data The greatest compilations of spontaneous ignition (autoignition) temperature are those of Mullins [AGARDOgraph No. 4 (1955)] and Zabetakis [US Bur. Mines Bull. 627 (1965)]. These data have been collated and are given with permission in this appendix. The largest compilation is that of Mullins, and consequently his general format of presenting the data is followed. There have been many methods for measuring ignition temperatures, and the results from different measurements have not necessarily been self-consistent. Mullins lists the various results, a reference to the technique used, and the reporting investigators. Since the various techniques have not been discussed in the text, these references have been omitted from the data that are reproduced here, and only the reported ignition temperatures are presented. All temperatures are reported in degrees centigrade. The delay period, where known, is in milliseconds and follows the temperature in parentheses. If no delay time appears, the spontaneous ignition time was either not specifiable, not specified, or determined by the manner shown in Fig. 7.1, and probably always in excess of 1000 ms. All data are for atmospheric pressure with the exception of the vitiated air data for acetylene and hydrogen, which are for 0.9 atm. When there was a significant difference between the values reported by Mullins and Zabetakis, the values given by Zabetakis were added to the reorganized Mullins compilation. This value is designated by the letter z following the temperature. When the original Mullins listing reported two or more values within 2 degrees of each other, only one value has been presented. If there are large differences between the reported values, the reader is urged to refer to Mullins and Zabetakis for the method and original source. Additional data are available in the Ignition Handbook by Babrauskas (Fire Science Publishers, Issaquah, WA, 2003) and from Kuchta [US Bur. Mines Bull. 680 (1985)].

721

722

Combustion

TABLE G1 Spontaneous Ignition Temperature Data Fuel

Oxygen

Air

Vitiated Air

Acetal

174

230

768 (20), 957 (1)

Acetaldehyde

140, 159

185, 275, 175z

869 (20), 1088 (1)

Acetic acid

570, 490

599, 550, 566, 465z

Acetic anhydride

361

392, 401

Acetone

568, 485

700, 727, 561, 538, 569, 465z

Acetanilide

546

Acetone cyanohydrin

688

Acetonitrile

1000 (20), 1059 (10)

Acetonylacetone

493, 340z,

Acetophenone

570z

Acetylchloride

390z

Acetylene

296

Acetyl oxide

(see Acetic anhydride)

Acrolein

305, 335

278, 235z

Acrylaldehyde

(see Acrolein)

Acrylonitrile

460

420z

Aldol

277, 248 348

Allylamine

712(10), 859 (1)

389

767 (20), 979 (1)

295

Allyl chloride

404

Allyl ether

200

Aminobenzene

(see Aniline)

o-Aminodiphenyl

487, 392 749 (10), 927 (1)

450z (see Monoethanolamine)

Aminoethylethanolamine

369

Ammonia

651

n-Amyl acetate

399, 360z, 378

i-Amyl acetate

379, 360z

n-Amyl alcohol

623 (20), 826 (1)

374

Allyl bromide

2-Aminoethanol

816 (20), 996 (1)

481

Adipic acid

Allyl alcohol

871 (20), 1046 (1)

390, 332

409, 427, 327, 300z

806 (20), 990 (1)

723

Appendix G: Spontaneous Ignition Temperature Data

TABLE G1 (continued) Fuel

Air

Vitiated Air

i-Amyl alcohol

518, 343, 353, 350z

818 (20), 1013 (1)

s-Amyl alcohol

343–385

t-Amyl alcohol

437

Amylbenzene

Oxygen

814 (20), 995 (1)

255

n-Amyl chloride

259

t-Amyl chloride

343

n-Amylene

273

n-Amyl ether

170z

i-Amyl ether

428

Amyl methyl ketone

311

Amyl nitrate

195z

524 (20), 798 (1)

Amyl nitrite

210z

496 (20), 910 (1)

i-Amyl nitrite

437 (10), 918 (1)

n-Amyl propionate Aniline

380z 530

770, 628, 530, 617, 593 (6000)

o-Anisidine

907 (20), 1065 (2)

787 (20), 1039 (1)

Anisole

560

744 (20), 1025 (1)

Anthracene

580

Antifebrin

(see Acetanilide)

Banana Oil

(see i-Amyl acetate)

Benzaldehyde

168

180, 192

744 (20), 936 (1)

Benzene

662, 690, 566

740, 656, 580, 645, 592 (42,000), 560z

814 (20), 1000 (1)

472, 540z

Benzene carbonal

(see Benzaldehyde)

Benzoic acid

475, 556

Benzyl acetate Benzyl alcohol Benzyl benzoate Benzyl cellosolve Benzyl chloride

373

573 588, 461

767 (20), 1019 (1)

502, 436

807 (20), 1007 (1)

480z (see Ethyleneglycolmonobenzyl ether) 627, 585z

(continues)

724

Combustion

TABLE G1 (continued) Fuel

Oxygen

Air

Benzyl ethanoate

(see Benzyl acetate)

Benzylethyl ether

496

Bicyclohexyl

245z

Biphenyl

577 (36,000), 540z

2-Biphenylamine

450z

Bromobenzene

688, 565z

1-Bromobutane

(see n-Butyl bromide)

Bromoethane

(see Ethyl bromide)

1,3-Butadiene

335

n-Butaldehyde

(see n-Butyraldehyde)

n-Butane

283

408, 430 (6000), 405z

i-Butane

319

462, 543, 477 (18,000)

1,3-Butandiol

Vitiated Air

858 (20), 1046 (1)

418

395z

2,3-Butanedione

(see Diacetyl)

1-Butanol

(see n-Butyl alcohol)

2-Butanol

(see s-Butyl alcohol)

2-Butanone

(see Methylethyl ketone)

2-Butenal

(see Crotonaldehyde)

1-Butene

310z

2-Butene

384 435, 325z

2-Butanol

(see s-Butyl alcohol)

2-Butoxyethanol

(see Ethyleneglycolmonobutyl ether)

n-Butyl acetate

423

793 (20), 1040 (1)

n-Butyl alcohol

385, 328

450, 503, 367, 359 (18,000)

809 (20), 993 (1)

i-Butyl alcohol

364

542, 441, 414

794 (20), 1010 (1)

s-Butyl alcohol

377

414, 405z

833 (20), 990 (1)

t-Butyl alcohol

460

478, 483, 480z

n-Butylamine

312

i-Butylamine

374

725

Appendix G: Spontaneous Ignition Temperature Data

TABLE G1 (continued) Fuel

Oxygen

Air

t-Butylamine

380z

n-Butylbenzene

412, 444 (6000)

i-Butylbenzene

428, 456 (12,000)

s-Butylbenzene

443, 420z, 447 (18,000)

t-Butylbenzene

448, 477 (72,000)

2-Butylbiphenyl

433 (12,000)

n-Butyl bromide

483, 316, 265z

Butyl carbinol

228

Butyl carbitol acetate

299 (see Ethyleneglycolmonobutyl ether)

n-Butyl chloride

460

α-Butylene

(see 1-Butene)

β-Butylene

(see 2-Butene)

γ-Butylene

(see 2-Methylpropene)

i-Butylene

(see 2-Methylpropene)

b-Butylene glycol

377

n-Butyl ether

194

n-Butyl formate

308

322

Butyl lactate

382

i-Butylmethyl ketone

459

n-Butyl nitrite

400 (4), 490 (1)

Butylphthalate

403

n-Butyl propionate

426

n-Butylstearate

355z

n-Butyraldehyde

779 (20), 1000 (1)

(see i-Amyl alcohol)

Butyl carbitol

Butyl cellosolve

Vitiated Air

206

813 (20), 1021 (1)

408, 230

i-Butyraldehyde

254

n-Butyric acid

552, 450z

Camphor

466

Carbon disulfide

107

149, 120, 125, 90z

610 (20), 842 (1)

Carbon monoxide

588

609, 651

758 (20), 848 (1) (continues)

726

Combustion

TABLE G1 (continued) Fuel

Oxygen

Castor oil Cellosolve

Air

Vitiated Air

449 (see Ethyleneglycolmonoethyl ether)

Cetane

235

Cetene

748 (20), 1036 (1)

o-Chloroaniline

885 (20), 1084 (2)

m-Chloroaniline

846 (20), 1080 (2)

Chlorobenzene

674, 640z

Chloroethane

(see Ethyl chloride)

2-Chloro-2-methyl chloride

318

343

3-Chloro-(trifluoromethyl) benzene

654

Creosote oil

336

o-Cresol

599

m-Cresol

626

836 (20), 1100 (2)

Crotonaldehyde

232

703 (20), 924 (1)

Cumene

467 (6000), 425z

802 (20), 985 (1)

pseudo-Cumene

770 (20), 1025 (2)

Cyanogen

850

Cyclohexadiene

360

Cyclohexane

325, 296

259, 270 (102,000), 245z

798 (20), 980 (1)

Cyclohexanol

350

300z

814 (20), 1030 (1)

Cyclohexanone

550

557, 453

816 (20), 1046 (1)

Cyclohexylamine

293

Cyclohexyl acetate

335z

Cyclohexene

325

Cyclopentadiene

510

Cyclopentane

385(6000)

Cyclopentanone

540

Cyclopropane

454

p-Cymene Decahydronaphthalene

781 (20), 972 (1)

498 466, 494, 445, 435z

280

262, 272 (18,000), 250z

807 (20), 1050 (1)

727

Appendix G: Spontaneous Ignition Temperature Data

TABLE G1 (continued) Fuel

Oxygen

Air

trans-Decahydronaphthalene

814 (20), 1002 (1)

Decalin

(see Decahydronaphthalene)

n-Decane

202

463, 425, 250, 210z, 232 (54,000), 236

1-Decanol

291

1-Decene

244 (78,000)

n-Decyl alcohol

793 (20), 960 (1)

Diacetone alcohol

603

Diacetyl (see Acetonylacetone)

Diallyl

330

Diallyl ether

(see Allyl ether)

Dibutyl ether

(see n-Butyl ether)

Dibutyl phthalate

(see Butyl phthalate)

Di-n-butyl tartrate

284

o-Dichlorobenzene

648 250

Dichloro-1(chlorotetrafluoroethyl)-4(trifluoromethyl) benzene 1,2-Dichloroethane

805 (20), 1065 (1) 748 (20), 930 (1)

1,2-Diacetylethane

1,2-Dichloro-n-butane

Vitiated Air

276 591

(see Ethylene dichloride)

Dichloroethylene

441, 458

738 (20), 1079 (1)

2,2’-Dichloroethyl ether

369

766 (20), 953 (1)

Dichloromethane

(see Methylene chloride)

1,2-Dichloropropane

(see Propylene dichloride)

Dicyclopentadiene

510

Di-n-decyl ether

217

Diesel fuel (41 cetane)

233z

Diesel fuel (55 cetane)

230z

Diesel fuel (60 cetane)

225z

Diesel fuel (68 cetane)

226z

Diethanolamine

662

823 (20), 1000 (2) (continues)

728

Combustion

TABLE G1 (continued) Fuel

Oxygen

Air

Vitiated Air

1,1-Diethoxyethane

(see Acetal)

Diethylamine

312

754 (20), 977 (1)

Diethylaniline

630z

762 (20), 965 (1)

1,2-Diethylbenzene

404 (6000)

1,3-Diethylbenzene

455 (12,000)

1,4-Diethylbenzene

430, 451 (12,000)

Diethylcellosolve

(see Ethyleneglycoldiethyl ether)

Dimethyl cyclohexane

240z

1,4-Diethylene dioxide

(see Dioxane)

Diethyl ether

(see Ethylether)

Diethylene glycol

229

Diethyleneglycolbenzoate-2ethylhexoate

340

Diethylene oxide

(see Dioxane)

Diethylenetriamine

399

Diethyl ketone

608, 450z

3,3-Diethylpentane

322, 290z

Diethyl peroxide

189

Diethyl sulfate

436

Dihexyl

(see Dodecane)

Di-n-hexyl ether 2,2’-Dihydroxyethylamine

200 (see Diethanolamine)

Diisobutylenes

470

Diisooctyladipate

366

Diisopropyl

(see 2,3-Dimethylbutane)

Diisopropylbenzene

449

Diisopropyl ether

443, 416, 500

Dimethylamine

346

371

2,2-Dimethylbutane

425, 440 (12,000)

2,3-Dimethyl-1-butene

298

820 (20), 1037 (1)

402

Dimethylaniline

2,3-Dimethylbutane

799 (20), 1064 (1)

420, 421(12,000) 369 (6000)

780 (20), 960 (1)

729

Appendix G: Spontaneous Ignition Temperature Data

TABLE G1 (continued) Fuel

Oxygen

Air

2,3-Dimethyl-2-butene

407 (6000)

Dimethylchloracetal

232

Dimethyl decalin

235z

2,4-Dimethyl-3-ethylpentane

390 (12,000), 510

Dimethyl ether

252

trans-Dimethylethylene

(see 2-Methyl propene)

Dimethyl formamide Dimethylglyoxal

350

445, 435z (see Diacetyl)

3,3-Dimethylheptane

441 (3600)

2,3-Dimethylhexane

438

Dimethylketone

Vitiated Air

(see Acetone)

2,3-Dimethyloctane

231 (72,000)

4,5-Dimethyloctane

388

2,3-Dimethylpentane

337, 338 (6000)

o-Dimethylphthalate

556

2,2-Dimethylpropane

450, 456 (3000)

Dimethyl sulfide

206

1,1-Dineopentylethane

500

1,1-Dineopentylethylene

455

Dioctylbenzenephosphonate

314

Di-n-octyl ether

210

Dioctylisooctenephosphonate

320

1,4-Dioxane

266, 179

Dipentene

237z

Diphenylamine

452, 635z

1,1-Diphenylbutane

462 (6000)

1,1-Diphenylethane

487 (6000)

Diphenylether

620z

Diphenylmethane

517 (18,000), 485z

Diphenyloxide

646 (12,000)

1,1-Diphenylpropane

466 (6000)

775 (20), 933 (1)

(continues)

730

Combustion

TABLE G1 (continued) Fuel

Oxygen

Air

Di-n-propyl ether

189

Divinyl ether

360

n-Dodecane

232, 205z

i-Dodecane

534, 500

1-Dodecanol

283

n-Eicosane

240

Vitiated Air

827 (20), 1010 (1)

Ethanal

(see Acetaldehyde)

Ethane

506z

Ethanol

(see Ethyl alcohol)

Ethene

(see Ethylene)

Ether

(see Ethylether)

p-Ethoxyaniline

(see p-Phenetidine)

2-Ethoxyethanol

(see Ethyleneglycolmonoethyl ether)

2-Ethoxylethanol acetate

(see Ethyleneglycolmonoethyl ether monoacetate)

Ethyl acetate Ethyl alcohol

425, 375

472, 515

610, 486

804 (20), 1063 (1)

558, 426, 365z

814 (20), 1030 (1)

Ethylamine (70% aqueous solution)

384

Ethylaniline

479

Ethylbenzene

468

460 (18,000), 553, 477, 430z

Ethylbenzoate

644

2-Ethylbiphenyl

449 (18,000)

Ethyl bromide

588, 511

2-Ethylbutane

273

2-Ethyl-1-butanol

(see i-Hexyl alcohol)

2-Ethyl-1-butene Ethyl-n-butyrate Ethyl caprate

785 (20), 966 (1)

883 (20), 1055 (2)

324 (6000) 351

612, 463 493

Ethyl-n-caproate

582

Ethyl-n-caprylate

571

Ethyl carbonate

809 (20), 991 (1)

782 (20), 1013 (1)

731

Appendix G: Spontaneous Ignition Temperature Data

TABLE G1 (continued) Fuel

Oxygen

Air

Ethyl chloride

468

516, 494

Ethylcyclobutane

211

Ethylcyclohexane

262, 264 (114,000)

Ethylcyclopentane

262

Ethylene

485

490, 543

Ethylene chlorhydrin

400

425

Ethylene dichloride Ethyleneglycol

Vitiated Air

413 500

522, 413, 400z

Ethyleneglycol diacetate

635

Ethyleneglycoldiethyl ether

208

Ethyleneglycolmonobenzyl ether

352

Ethyleneglycolmonobutyl ether

244

792 (20), 964 (1)

Ethyleneglycolmonoethyl ether

238, 350z

790 (20), 954 (1)

Ethyleneglycolmonoethyl ether acetate

379

Ethyleneglycolmonoethyl ether monoacetate

774 (20), 960 (1)

Ethyleneglycolmonomethyl ether

288, 382

Ethylene imine

322

Ethylene oxide

429

Ethyl ether

178, 182

780 (20), 933 (1)

343, 491, 186, 193, 160z

794 (20), 947 (1)

Ethyl formate

577, 455z

768 (20), 956 (1)

Bis (2-Ethylhexyl)-adipate

262

Ethyl lactate

400

Ethyl malonate

541

Ethyl mercaptan

261

Ethyl methyl ether

(see Methyl ethyl ether)

Ethyl methyl ketone

(see Methyl ethyl ketone)

1-Ethyl naphthalene

299

481 (6000) (continues)

732

Combustion

TABLE G1 (continued) Fuel

Oxygen

Air

Ethyl nitrate

426 (20), 562 (1)

Ethyl nitrite

90

3-Ethyloctane

235

4-Ethyloctane

237 (54,000)

Ethyloleate

353

Ethyl oxalate 388

Ethyl pelargonate

524 440

602, 476, 440z

Ethylpropyl ketone

575

Ethyl-n-valerianate

590

Formaldehyde (37% solution)

430

Formamide

969 (20), 1032 (10)

Formic acid

504

Furan 2-Furancarbonal

783 (20), 982 (1) (see Furfuraldehyde)

Furfuraldehyde

391

696 (20), 880 (1)

391, 491

775 (20), 944 (1)

Furfuran

(see Furan)

Furfuryl alcohol

364

Fusel oil

(see n-Amyl alcohol)

Gas oil

270

336

Gasoline (100/130)

440z

Gasoline (115/145)

470z

Glycerine

370z

Glycerol

580 (20), 833 (1)

742 (10), 880 (1)

Ethyl palmitate

Ethyl propionate

Vitiated Air

414, 320

Glyceryl triacetate

500, 523, 393 433

Glycol

(see Ethylene glycol)

n-Heptane

300, 214, 209z

451, 230, 233, 250, 247 (30,000), 215z, 223z

n-Heptanoic acid

523

1-Heptene

332, 263 (66,000)

806 (20), 950 (1)

733

Appendix G: Spontaneous Ignition Temperature Data

TABLE G1 (continued) Fuel

Oxygen

α-n-Heptylene

(see 1-Heptene)

Air

Hexachlorobutadiene

618 (6000)

Hexachlorodiphenyl oxide

628, 600

n-Hexadecane

230 (66,000), 232, 205z

i-Hexadecane

484

1-Hexadecene

240 (78,000), 253

Hexahydrobenzene

(see Cyclohexane)

Hexahydrophenol

(see Cyclohexanol)

Hexamethylbenzene

375

Hexamethylene

(see Cyclohexane)

n-Hexane

296, 225z

487, 520, 248, 261, 225z, 261 (30,000)

i-Hexane

268, 284

2,5-Hexanedione

(see Acetonylacetone)

1-Hexene

(see i-Butylmethyl ketone)

n-Hexyl alcohol

300

801 (20), 970 (1)

i-Hexyl alcohol

800 (20), 963 (1) 325

n-Hexylether

185z

Hydrazine

270

Hydrocyanic acid

538

Hydrogen

828 (20), 1015 (1)

272 (72,000)

Hexone

Hexylene

Vitiated Air

560

572, 400z

Hydrogen sulfide

220

292

Hydroquinone

630

2-Hydroxyethylamine

(see Monoethanolamine)

4-Hydroxy-4-methyl-2pentanone

(see Diacetone alcohol)

α-Hydroxytoluene

(see Benzyl alcohol)

m-Hydroxytoluene

(see m-Cresol)

Isophorone

322

610 (20), 700 (1)

462 (continues)

734

Combustion

TABLE G1 (continued) Fuel

Oxygen

Isoprene

440

Air

JP-1

228z

JP-3

238z

JP-4

240z

JP-6

230z

Kerosine

270

295, 254, 210z, 249 (66,000)

Ketohexahydrobenzene

(see Cyclohexanone)

dl-Limonene

263 (30,000)

Linseed oil

438

Mesitylene

621

Mesityl oxide

806 (20), 998 (1)

344

823 (20), 1037 (1)

632, 537, 540z

961 (20), 1050 (10)

Methanamide

(see Formamide)

Methane

556

Methanol

(see Methyl alcohol)

Methone

Vitiated Air

506

o-Methoxyaniline

(see o-Anisidine)

2-Methoxyethanol

(see Ethyleneglycolmonomethyl ether)

Methyl acetate

654, 502

816 (20), 1028 (2) 820 (20), 1040 (1)

Methyl alcohol

555, 500, 461

574, 470, 464, 385z

Methylamine

400

430

Methylaniline

(see Toluidine)

2-Methylbiphenyl

502 (12,000)

Methyl bromide

537

2-Methylbutane

294

420, 427 (6000)

2-Methyl-2-butanol

(see t-Amyl alcohol)

3-Methyl-1-butanol

(see i-Amyl alcohol)

3-Methyl-1-butene

374 (6000)

1-Methyl-2-t-butylcyclohexane

314 (12,000)

1-Methyl-3-t-butylcyclohexane (high boiling isomer)

304 (12,000)

1-Methyl-3-t-butylcyclohexane (low boiling isomer)

291 (24,000)

735

Appendix G: Spontaneous Ignition Temperature Data

TABLE G1 (continued) Fuel

Oxygen

Methylbutyl ketone Methyl cellosolve

(see Ethyleneglycolmonomethyl ether) 632

Methyl cyanide

(see Acetonitrile)

Methylcyclohexane

285

Methylcyclohexanone

265 (108,000), 250z 598

329

Methylcyclopentadiene

323 (6000)

231 (see Methylene chloride)

1-Methyl-3,5-diethylbenzene

461 (12,000)

Methylene chloride

606

Methyl ether

(see Dimethyl ether)

642, 662, 615z

1-Methyl-2-ethylbenzene

447 (18,000)

1-Methyl-3-ethylbenzene

485 (18,000)

1-Methyl-4-ethylbenzene

483 (12,000)

Methyl ethyl ether

190

Methyl ethyl ketone

514, 505

Methylethylketone peroxide

390z

2-Methyl-3-ethylpentane

461

Methylformate

236, 449, 465z

Methylheptenone

534

Methylcyclohexanol

295z

Methylhexyl ketone

572

Methylisopropylcarbinol

902 (20), 1085 (2)

804 (20), 975 (1)

797 (20), 1026 (2)

(see s-Amyl alcohol)

Methyl lactate

385

1-Methylnaphthalene

566, 547 (24,000), 553, 530z

2-Methylnonane

214 (102,000)

2-Methylpropanal

812 (20), 1013 (1)

445z

2-Methyldecane Methylenedichloride

Vitiated Air

533

Methyl chloride

Methylcyclopentane

Air

(see i-Butyraldehyde)

2-Methyloctane

227 (66,000)

3-Methyloctane

228 (60,000) (continues)

736

Combustion

TABLE G1 (continued) Fuel

Oxygen

4-Methyloctane 2-Methylpentane

Air 232 (6000)

275

306, 307 (6000)

3-Methylpentane

304 (12,000)

2-Methyl-1-pentene

306 (6000)

4-Methyl-1-pentene

304 (12,000)

4-Methyl-3-pentene-2-one

(see Mesityl oxide)

2-Methyl-1-propanol

(see i-Butyl alcohol)

α-Methylpropyl alcohol

(see s-Butyl alcohol)

2-Methyl propane

(see i-Butane)

2-Methylpropene

465

Methylpropionate

469

Methyl-n-propyl ketone

505

2-Methylpyridine

454 (see Tetrahydrosylvan)

Methylstyrene

495z

Monoethanolamine

780 (20), 1006 (1)

Monoisopropyl bicyclohexyl

230z

2-Monoisopropyl biphenyl

435z

Monoisopropylxylenes Naphthalene

832 (20), 1020 (1)

(see a-Picoline)

Methylsalicylate 2-Methyltetrahydrofuran

Vitiated Air

798 (20), 1040 (1) 630, 560

Neatsfoot oil

587, 568, 526z 442

Neohexane

(see 2,2-Dimethylbutane)

Nicotine

235

244

Nitrobenzene

556, 482

706 (20), 884 (1)

Nitroethane

415

623 (20), 762 (1)

Nitroglycerol

270

Nitromethane

419

1-Nitropropane

417

2-Nitropropane

428

o-Nitrotoluene

684 (20), 784 (1)

672 (20), 837 (1)

737

Appendix G: Spontaneous Ignition Temperature Data

TABLE G1 (continued) Fuel

Oxygen

Air

m-Nitrotoluene

Vitiated Air 711 (20), 911 (1)

n-Nonadecane

237

n-Nonane

285, 234 (66,000), 205z

n-Octadecane

235

1-Octadecene

251

Octadecyl alcohol

270

Octahydroanthracene

315

n-Octane

208

i-Octane

(see 2,2,4-Trimethylpentane)

458, 218, 240 (54,000)

1-Octene

256 (72,000)

Octyleneglycol

335

Olive oil

441

Oxalic acid

640

Oxybutyricaldehyde

(see Aldol)

Palm oil

343

Paraldehyde

541, 242

Pentamethylene glycol

335z

n-Pentane

300, 258

i-Pentane

(see 2-Methylbutane)

1-Pentene

846 (20), 1064 (1)

579, 290, 284 (24,000), 260z

298 (18,000)

1-Pentanol

(see n-Amyl alcohol)

2-Pentanone

(see Methylpropyl ketone)

γ-Pentylene oxide

(see Tetrahydrosylvan)

Perfluorodimethylcyclohexane

651 (6000)

p-Phenetidine

822 (20), 1004 (1)

Phenol

574, 500

715

Phenylamine

(see Aniline)

Phenylaniline

(see Diphenylamine)

Phenylbromide

(see Bromobenzene)

Phenyl carbinol

(see Benzyl alcohol) (continues)

738

Combustion

TABLE G1 (continued) Fuel

Oxygen

Air

Phenyl chloride

(see Chlorobenzene)

n-Phenyldiethylamine

(see Diethylaniline)

Phenylethane

(see Ethylbenzene)

Phenylethylene

(see Styrene)

Phenylmethyl ether

(see Anisole)

Phenylmethyl ketone

(see Acetophenone)

Phosphorus (yellow)

30

Phosphorus (red)

260

Phosphorus sesquisulfide

100

Phthalic anhydride

584, 570z

α-Picoline

538, 500z

Picric acid

300

Pinene

275

1,2-Propandiol

263 (60,000)

Vitiated Air

846 (20), 1037 (2)

797 (20), 1039 (1)

410z

Propane

468

493, 450z, 466, 504 (6000)

1-Propanol

(see n-Propyl alcohol)

2-Propanol

(see i-Propyl alcohol)

2-Propanone

(see Acetone)

Propenal

(see Acrolein)

Propene

423z

Propen-1-ol

(see Allyl alcohol)

Propene oxide

(see Propylene oxide)

458

Propionaldehyde

419

n-Propionic acid

596

n-Propyl acetate

388

662, 450

i-Propyl acetate

448

572, 460, 476

n-Propyl alcohol

445, 370, 328

505, 540, 439, 433

811 (20), 1007 (1)

i-Propyl alcohol

512

590, 620, 456

811 (20), 1050 (1)

n-Propylamine

318

i-Propylamine

402

n-Propylbenzene

456 (12,000)

828 (20), 1060 (2)

739

Appendix G: Spontaneous Ignition Temperature Data

TABLE G1 (continued) Fuel

Oxygen

i-Propylbenzene

(see Cumene)

2-Propylbiphenyl Propyl bromide

Air

452 (18,000), 440z 255

490

Propyl chloride

520

i-Propyl chloride

593

Propylcyclopentane

285

Propylene

(see Propene)

Propylenealdehyde

(see Crotonaldehyde)

Propylene dichloride Propylene glycol

557 392

748 (10), 870 (1) (see Diisopropyl ether)

n-Propyl formate

455

n-Propyl nitrate

175z

i-Propyl formate

485

4-i-Propylheptane

288

p-i-Propyltoluene

(see p-Cymene)

Pseudocumene

(see pseudo-Cumene)

Pulegon

426

Pyridine Pyrogallol

482, 574

480 575

Rape seed oil

446

Rosin oil

342

Salicylicaldehyde

772 (20), 1015 (1)

Soya bean oil

445

Stearic acid

250

395

Styrene

450

490

Sugar

378

385

Sulfur

745 (20), 1014 (1)

510

Quinoline Quinone

790 (20), 1012 (2)

421

Propylene oxide i-Propyl ether

Vitiated Air

777 (20), 1065 (1)

232 (continues)

740

Combustion

TABLE G1 (continued) Fuel

Oxygen

Air

Tannic acid

527

Tartaric acid

428

p-Tephenyl

535z

Tetraaryl salicylate

577 (6000)

n-Tetradecane

232, 200z

1-Tetradecene

239 (66,000), 255

Vitiated Air

1,2,3,4-Tetrahydrobenzene

(see Cyclohexene)

Tetrahydrofurfuryl alcohol

273

282

793 (20), 980 (1)

Tetrahydronaphthalene

420

423 (6000), 385z

819 (20), 1030 (1)

Tetrahydrosylvan

794 (20), 1025 (1)

Tetraisobutylene

415

n-Tetradecane

200z

Tetralin

(see Tetrahydronaphthalene)

Tetramethylbenzene

791 (20), 1030 (1)

Tetramethylene glycol

390z

2,2,3,3-Tetramethylpentane

430, 452 (42,000), 516

2,3,3,4-Tetramethylpentane

437 (24,000), 514

Toluene

552, 640, 516

810, 633, 552, 540, 480z, 568 (48,000), 635

830 (20), 1057 (1)

o-Toluidine

537, 482

832 (20), 1040 (3)

m-Toluidine

580

846 (10), 1062 (2)

p-Toluidine

482

Tributyl citrate

368

Trichloroethane Trichloroethylene

500z 419

463, 420z

Trichloro-1-(pentafluoroethyl)-4(trifluoromethyl) benzene

568

Tricresyl phosphate

600

Triethylene glycol

244

771 (10), 950(1)

371

Triethylenetetramine

338

Triisobutylenes

413

1,2,3-Trimethylbenzene

479 (24,000), 510

1,2,4-Trimethylbenzene

521 (24,000), 528

789 (20), 1060 (1)

741

Appendix G: Spontaneous Ignition Temperature Data

TABLE G1 (continued) Fuel

Oxygen

Air

1,3,5-Trimethylbenzene

559 (48,000), 577

2,2,3-Trimethylbutane

454 (18,000), 420z

2,3,3-Trimethyl-1-butene

383 (12,000)

Trimethylene glycol

400z

2,5,5-Trimethylheptane

485

2,2,3-Trimethylpentane

436 (24,000)

2,2,4-Trimethylpentane

283

561, 434, 447 (12,000), 515, 415z

2,3,3-Trimethylpentane

430 (12,000)

2,3,4-Trimethyl-1-pentene

257 (12,000)

2,4,4-Trimethyl-1-pentene (cf. Diisobutylenes)

420 (12,000)

2,4,4-Trimethyl-2-pentene (cf. Diisobutylenes)

308 (30,000)

3,4,4-Trimethyl-2-pentene

330 (24,000)

2,4,6-Trimethyl-1,3,5-trioxane

(see Paraldehyde)

Trinitrophenol

(see Picric acid)

Trioxane Tritolyl phosphate

786 (20), 996 (2)

424, 414 (see Tricresyl phosphate)

Tung oil

457

Turkey red oil

445

Turpentine

252, 255

Vinyl acetate

427

Vinylcyclohexene

269

Vinyl ether

Vitiated Air

(see Divinyl ether)

Vinylethyl ether

201

Vinyl-2-ethylhexyl ether

201

Vinylisopropyl ether

272

o-Xylene

496, 465z, 501 (30,000), 551

m-Xylene

563 (54,000), 652, 530z

p-Xylene

618, 564 (42,000), 657, 530z

Zinc stearate

421

780 (20), 996 (1)

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Appendix H

Minimum Spark Ignition Energies and Quenching Distances Most of the data presented in Table H1 are taken from Calcote et al. [Ind. Eng. Chem. 44, 2656 (1952)]. Additional data by Blanc et al. [Proceedings of the Combustion Institute 3, 363 (1949)] and Meltzer [NACA RM E53H31 (1953) and NACA RM E52 F27 (1952)] given in the table are designated by the letters B and M, respectively, in parentheses. Since the values for the least minimum ignition energy by Meltzer were extrapolated from low pressures, these values were not given if values by Calcote et al., or Blanc et al. were available. The column labeled plain contains data from 1/8-inch rod electrodes and that labeled flange contains data for the negative electrode flanged and the other electrode a 1/8 inch rod. Values in parentheses are taken from a Calcote et al. correlation between the two different types of electrode sets. Quenching distances can be obtained from the data presented in Table H1 and the correlation given as Fig. 7.5. It is interesting to note that the most stable fuels have the least minimum ignition energy in the region of 0.2 mJ.

743

744

Combustion

TABLE H1 Minimum Spark Ignition Energy Data for Fuels in Air at 1 atm Pressure

Minimum ignition energy (φ  1), 104 J

Least minimum ignition energy, 104 J

φ Value for least minimum ignition energy

1.0

Fuel

Plain

Flange

Acetaldehyde

3.76

(5.7)

Acetone

11.5

(21.5)

Acetylene

0.2

0.3

0.2

Acrolein

(1.37)

1.75

1.6 (M)

Allyl chloride

7.75

13.5

Benzene

5.5

(9.1), 7.8 (B)

2.25 (B)

1.8

1,3-Butadiene

(1.75)

2.35

1.25

1.4

n-Butane

7.6 (M)

7.0 (B)

2.6 (B)

1.5

Butane, 2,2-dimethyl-

16.4

(33)

2.5 (M)

1.4

Butane, 2-methyl-(Isopentane)

7.0

9.6

2.1 (M)

1.3

n-Butyl chloride

(12.4)

23.5

Carbon disulfide

0.15

0.39

Cyclohexane

13.8

(26.5), 10 (B)

Cyclohexane, methyl-

1.2 2.23 (M)

1.6

2.7 (M)

1.8

Cyclohexene

(5.25)

8.6

Cyclopentadiene

6.7

(11.4)

Cyclopentane

(5.4)

8.3

Cyclopropane

2.4

(3.4), 4.0 (B)

1.8 (B)

1.1

Diethylether

4.9

(7.9), 5.3 (B)

1.9 (B)

1.5

Dihydropyran

(3.65)

5.6

Diisopropyl ether

11.4

(21.4)

Dimethoxymethane

4.2

(6.6)

Dimethyl ether

(3.3), 2.9

4.5

Dimethyl sulfide

(4.8)

7.6

Di-t-butyl peroxide

(4.1)

6.5

745

Appendix H: Minimum Spark Ignition Energies and Quenching distances

TABLE H1 (continued) φ Value for least minimum ignition energy

Fuel

Plain

Flange

Least minimum ignition energy, 104 J

Ethane

(2.85)

4.2, 3.2 (B)

2.4 (B)

1.2

Ethene

0.96

(1.14)

1.24 (M)

1.1

Ethylacetate

14.2

(28)

4.8

1.2

Ethylamine

24

52

Ethylene oxide

0.87

1.05

0.62

1.3

Ethylenimine

4.8

(7.8)

Furan

2.25

(3.28)

Furan, tetrahydro-

5.4 (M)

Furan, thio(Thiophene)

(3.9)

6

n-Heptane

7.0

11.5, 11 (B)

2.4 (B)

1.8

1-Heptyne

(5.6)

9.31

Hexane

9.5 (M)

9.7 (B)

2.48 (B)

1.7

Hydrogen

0.2

0.3

0.18

0.8

Hydrogen sulfide

(0.68)

0.77

Isopropyl alcohol

6.5

(11.1)

Isopropylamine

(20)

41

Isopropyl chloride

(15.5)

31

Isopropyl mercaptan

(5.3)

8.7

Methane

4.7, 3.3 (M)

(7.1), 3.3 (B)

2.8 (B)

0.9

Methanol

2.15

(3.0)

1.4 (M)

1.3

Methyl acetylene

1.52

(2)

1.2

1.4

Methyl ethyl ketone

(6.8), 5.3

11

2.8

1.4

Methyl formate

(4.0)

6.2

n-Pentane

(5.1), 4.9

8.2

2.2 (M)

1.3

29

2.8 (M)

Minimum ignition energy (φ  1), 104 J

n-Pentane, 2,4,4-trimethyl-(Iso-octane) 13.5

(continues)

746

Combustion

TABLE H1 (continued)

Minimum ignition energy (φ  1), 104 J

Least minimum ignition energy, 104 J

φ Value for least minimum ignition energy

Fuel

Plain

Flange

1-Pentene, 2,4,4trimethyl(Diisobutylene)

(9.6)

17.5

2-Pentene

(5.1), 4.7

8.2

1.8 (M)

1.6

Propane

3.05

5, 4.0 (B)

2.5

1.3

Propane, 2,2-dimethyl(Neopentane)

15.7

(31)

Propane, 2-methyl-(Isobutane)

(5.2)

8.5

Propene

2.82

(4.18), 4.1

Propionaldehyde

(3.25)

4.9

n-Propyl chloride

(10.8)

20

Propylene oxide

1.9

2.1

1.4

1.4

Tetrahydropyran

12.1

(23)

2.2 (M)

1.7

Triethylamine

(7.5), 11.5

13

Triptane

10

(18.2)

Vinyl acetate

(7.0)

12.0

Vinyl acetylene

0.822

(0.95)

Appendix I

Programs for Combustion Kinetics The increase in thermochemical and kinetic databases and the development of fast and affordable personal computers and workstations have enabled the use of many programs for studying combustion kinetics problems. In this appendix, a listing of some of the available programs for studying combustion phenomena is provided. Some of these programs, as well as many others, may be found and retrieved from the Internet by performing a search using keywords such as “chemical kinetics computer codes” or by visiting the homepage of various laboratories and universities, for example, the homepage of the National Institute of Standards and Technology (NIST).

A. THERMOCHEMICAL PARAMETERS THERM: “Thermodynamic Property Estimation for Radicals and Molecules,” Edward R. Ritter and Joseph Bozzelli, Int. J. Chem. Kinet. 23, 767–778, (1991). A computer program for IBM PC and compatibles for estimating, editing, and entering thermodynamic property data for gas-phase radicals and molecules using Benson’s group additivity method. RADICALC: Bozzelli, J. W. and Ritter, E. R. Chemical and Physical Processes in Combustion, p. 453. The Combustion Institute, Pittsburgh, PA, 1993. A computer code to calculate entropy and heat capacity contributions to transition states and radical species from changes in vibrational frequencies, barriers, moments of inertia, and internal rotations. FITDAT: Kee, R. J., Rupley, F. and Miller, J. A. Sandia National Laboratories, Livermore, CA 94550. A Fortran computer code (fitdat.f) that is part of the CHEMKIN package for fitting of species thermodynamic data (cp, h, s) to polynomials in NASA format for usage in computer programs.

B. KINETIC PARAMETERS UNIMOL: “Calculation of Rate Coefficients for Unimolecular and Recombination Reactions,” Gilbert, R. G., Jordan, M. J. T. and Smith, S. C. Department of Theoretical Chemistry, Sydney, Australia (1990). Fortran computer code for calculating the pressure and temperature dependence of unimolecular and 747

748

Combustion

recombination (association) rate coefficients. Theory based on RRKM and numerical solution of the master equation. See “Theory of Unimolecular and Recombination Reactions,” by R. G. Gilbert and S. C. Smith, Blackwell Scientific Publications, Oxford, 1990. CHEMACT: “A Computer Code to Estimate Rate Constants for Chemically-Activated Reactions,” Dean, A. M., Bozzelli, J. W. and Ritter, E. R. Combust. Sci. Tech. 80, 63–85 (1991). A computer code based on the QRRK treatment of chemical activation reactions to estimate apparent rate constants for the various channels that can result in addition, recombination, and insertion reactions. NIST: Chemical Kinetics Database, Mallard, N. G., Westley, F., Herron, J. T., Hampson, R. F. and Frizzell, D. H. NIST, NIST Standard Reference Data, Gaithersburg, MD, 1993. A computer program for IBM PC and compatibles for reviewing kinetic data by reactant, product, author, and citation searches and for comparing existing data with newly evaluated data. CHEMRATE: A Calculational Data Base for Unimolecular Reaction, Mokrushin, V. and Tsang, W. NIST, Gaithersburg, MD, http://www.nist.gov/ kinetics/chemrate/chemrate.html. CHEMRATE is a program that contains databases containing experimental results on unimolecular reactions, information pertaining to transition state and molecular structures necessary for the calculation of high pressure rate constants and thermal functions, respectively. A master equation solver is included so that rate constants for unimolecular reactions in the energy transfer region and chemical activation processes under steady and non-steady state conditions can be evaluated based on RRKM theory.

C. TRANSPORT PARAMETERS TRANFIT: “A Fortran Computer Code Package for the Evolution of Gas-Phase Multicomponent Transport Properties,” Kee, R. J., Dixon-Lewis, G., Warnatz, J., Coltrin, M. E. and Miller, J. A. Sandia National Laboratories, Livermore, CA, Sandia Report SAND86-8246, 1986. TRANFIT is a Fortran computer code (tranlib.f, tranfit.f, and trandat.f) that allows for the evaluation and polynomial fitting of gas-phase multicomponent viscosities, thermal conductivities, and thermal diffusion coefficients.

D. REACTION MECHANISMS CHEMKIN-II: “A Fortran Chemical Kinetics Package for the Analysis of GasPhase Chemical Kinetics,” Kee, R. J., Rupley, F. M. and Miller, J. A. Sandia National Laboratories, Livermore, CA, Sandia Report SAND89-8009, 1989. A Fortran computer program (cklib.f and ckinterp.f) designed to facilitate the formation, solution, and interpretation of problems involving elementary gas-phase chemical kinetics. The software consists of two programs: an Interpreter and a Gas-Phase Subroutine Library. The Interpreter program

Appendix I: Programs for Combustion Kinetics

749

converts a user-supplied text file of a reaction mechanism into vectorized binary output that forms a link with the gas-phase library. The two files can then be used in conjunction with user-supplied driver and solution codes for different kinetic related problems. See for example the PREMIX, SENKIN, PSR, and EQUIL codes described below. The software was upgraded to CHEMKIN-III, which was the last version of CHEMKIN distributed from Sandia National Laboratories (http://www.ca.sandia.gov/chemkin/). CHEMKIN is now maintained and distributed by Reaction Design, Inc., which is a software company licensed by Sandia National Laboratories. CHEMKIN 4.1 is the latest commercial version of the CHEMKIN software suite from Reaction Design. The software suite has all the application modules of CHEMKIN II and III (such as SURFACE CHEMKIN, EQUIL, SENKIN, PSR, and PREMIX), and has been extended to include many more. Refer to the website http://www.reactiondesign.com/lobby/open/index.html for more information. SURFACE CHEMKIN: “A Fortran Package for Analyzing Heterogeneous Chemical Kinetics at a Solid-Surface — Gas-Phase Interface,” Coltrin, M. E., Kee, R. J. and Rupley, F. M. Sandia National Laboratories, Livermore, CA, Sandia Report SAND90-8003C, 1990. A Fortran computer program (sklib.f and skinterp.f) used with CHEMKIN-II/III for the formation, solution, and interpretation of problems involving elementary heterogeneous and gas-phase chemical kinetics in the presence of a solid surface. The user format is similar to CHEMKIN-II/III. SURFTHERM: Coltrin, M. E. and Moffat, H. K. Sandia National Laboratories. SURFTHERM is a Fortran program (surftherm.f) that is used in combination with CHEMKIN (and SURFACE CHEMKIN) to aid in the development and analysis of chemical mechanisms by presenting in tabular form detailed information about the temperature and pressure dependence of chemical reaction rate constants and their reverse rate constants, reaction equilibrium constants, reaction thermochemistry, chemical species thermochemistry, and transport properties. CHEMKIN REAL-GAS: “A Fortran Package for Analysis of Thermodynamic Properties and Chemical Kinetics in Nonideal Systems,” Schmitt, R. G., Butler, P. B. and French, N. B. The University of Iowa, Iowa City, IA. Report UIME PBB 93-006, 1993. A Fortran program (rglib.f and rginterp.f) used in connection with CHEMKIN-II that incorporates several real-gas equations of state into kinetic and thermodynamic calculations. The real-gas equations of state provided include the van der Waals, Redlich-Kwong, Soave, Peng-Robinson, Becker-Kistiakowsky-Wilson, and Nobel-Abel. RMG-Reaction Mechanism Generation: “Automatic Reaction Network Generation using RMG for Steam Cracking of n-Hexane,” Van Geem, K., Reyniers, M. F., Marin, G., Song, J., Matheu, D. M. and Green, W. H. AIChE J. 52(2), 718–730 (2006) and “Building Robust Chemical Reaction Mechanisms: Next Generation Automatic Model Construction Software,” J. Song, PhD Thesis, MIT, 2004. The MIT Reaction Mechanism Generation (RMG) software package

750

Combustion

is a program that combines an extensive functional-group tree database with kinetic rate estimation and thermodynamic estimation parameters to generate a complete chemical kinetic model for a given chemical mixture. For more information, refer to the website http://web.mit.edu/cfgold/www/RMG/index.html. CHEMClean and CHEMDiffs: “The Comparison of Detailed Chemical Kinetic Mechanisms: Application to the Combustion of Methane,” Rolland, S. and Simmie, J. M. Int. J. Chem. Kinet. 36(9), 467–471, (2004). These programs may be used with CHEMKIN to (1) clean up an input mechanism file and (2) to compare two ‘clean’ mechanisms. Refer to the website http://www. nuigalway.ie/chem/c3/software.htm for more information. CHEMRev: “The Comparison of Detailed Chemical Kinetic Mechanisms; Forward Versus Reverse Rates with CHEMRev,” Rolland, S. and Simmie, J. M. Int. J. Chem. Kinet. 37(3), 119–125 (2005). This program makes use of CHEMKIN input files and computes the reverse rate constant, k(r), from the forward rate constant and the equilibrium constant at a specific temperature and the corresponding Arrhenius equation is statistically fitted, either over a usersupplied temperature range or, else over temperatures defined by the range of temperatures in the thermodynamic database for the relevant species. Refer to the website http://www.nuigalway.ie/chem/c3/software.htm for more information. CHEMThermo: “Automatic Comparison of Thermodynamic Data for Species in Detailed Chemical Kinetic Modeling,” Simmie, J. M., Rolland, S. and Ryder, E. Int. J. Chem. Kinet. 37(6), 341–345, (2005). CHEMThermo analyses the differences between two thermodynamic databases in CHEMKIN format, calculates the specific heat (Cp), the enthalpy (Ho), and the entropy (So) of a species at any given temperature, and compares the values of Cp, Ho, So at three different temperatures, for the species in common. Refer to the website http://www.nuigalway.ie/chem/c3/software.htm for more information. MECHMOD: A utility program written by Turányi, T. (Eötvös University, Budapest, Hungary) that manipulates reaction mechanisms to convert rate parameters from one unit to another, to calculate reverse rate parameters from the forward rate constant parameters and thermodynamic data, or to systematically eliminate select species from the mechanism. Thermodynamic data can be printed at the beginning of the mechanism, and the room-temperature heat of formation and entropy data may be modified in the NASA polynomials. MECHMOD requires the usage of either CHEMKIN-II or CHEMKIN-III software. Details of the software may be obtained at either of two websites http://www.chem.leeds.ac.uk/Combustion/Combustion.html or http://garfield. chem.elte.hu/Combustion/Combustion.html.

E. THERMODYNAMIC EQUILIBRIUM CEA: Chemical Equilibrium with Applications. “Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Analysis,” Gordon, S. and McBride, B. J. NASA Lewis Research Center, NASA Report

Appendix I: Programs for Combustion Kinetics

751

NASA RP-1311. NASA, Washington, DC, October, 1994. “Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Applications II. User’s Manual and Program Description,” Gordon, S. and McBride, B. J. NASA Lewis Research Center, NASA Report NASA RP-1311-P2. NASA, Washington, DC, June 1996. CEA calculates chemical equilibrium compositions and properties of complex mixtures. Applications include assigned thermodynamic states, theoretical rocket performance, Chapman–Jouguet detonations, and shock-tube parameters for incident and reflected shocks. CEA represents the latest in a number of computer programs that have been developed at the NASA Lewis (now Glenn) Research Center during the last 45 years. Associated with the program are independent databases with transport and thermodynamic properties of individual species. Over 2000 species are contained in the thermodynamic database. The program is written in ANSI standard Fortran and can operate on various platforms. Refer to the website http://www.grc.nasa. gov/WWW/CEAWeb/ for more information. CEC: “Computer Program for Calculation of Complex Chemical Equilibrium Compositions, Rocket Performance, Incident and Reflected Shocks, and Chapman–Jouguet Detonations,” S. Gordon and B.J. McBride, NASA Lewis Research Center, NASA Report NASA SP-273. NASA, Washington, DC. A Fortran computer program preceding CEA for calculating (1) chemical equilibrium for assigned thermodynamic states (T,P), (H,P), (S,P), (T,V), (U,V), or (S,V), (2) theoretical rocket performance for both equilibrium and frozen compositions during expansion, (3) incident and reflected shock properties, and (4) Chapman–Jouguet detonation properties. The approach is based on minimization of free energy and considers condensed phase as well as gaseous species. A useful program for obtaining thermodynamic input is given in the report “Computer Program for Calculating and Fitting Thermodynamic Functions,” McBride, B. J., and Gordon, S. NASA Lewis Research Center, NASA RP-1271(1992), NASA, Washington, DC. STANJAN: “The Element Potential Method for Chemical Equilibrium Analysis: Implementation in the Interactive Program STANJAN,” W.C. Reynolds, Thermosciences Division, Department of Mechanical Engineering, Stanford University, Stanford, CA, 1986. A computer program for IBM PC and compatibles for making chemical equilibrium calculations in an interactive environment. The equilibrium calculations use a version of the method of element potentials in which exact equations for the gas-phase mole fractions are derived in terms of Lagrange multipliers associated with the atomic constraints. The Lagrange multipliers (the “element potentials”) and the total number of moles are adjusted to meet the constraints and to render the sum of mole fractions unity. If condensed phases are present, their populations also are adjusted to achieve phase equilibrium. However, the condensed-phase species need not be present in the gas-phase, and this enables the method to deal with problems in which the gas-phase mole fraction of a condensed-phase species is extremely low, as with the formation of carbon particulates.

752

Combustion

EQUIL: A.E. Lutz and F. Rupley, Sandia National Laboratories, Livermore, CA. A Fortran computer program (eqlib.f) for calculating chemical equilibrium using a modified solution procedure of STANJAN (stanlib.f, W.C. Reynolds, Stanford U.) and CHEMKIN data files for input. For the most recent versions, refer to the Reaction Design website http://www.reactiondesign.com/lobby/ open/index.html. CANTERA: Object-Oriented Software for Reacting Flows. Cantera is an open-source, object-oriented software package for problems involving chemically reacting flows (http://www.cantera.org/index.html). Capabilities include multiphase chemical equilibrium, thermodynamic and transport properties, homogeneous and heterogeneous kinetics, reactor networks, constant volume and pressure explosions, stirred reactors, one-dimensional flames, reaction path diagrams, and interfaces for MATLAB, Python, C, and Fortran. The program was developed initially by Goodwin, D. G. with significant contributions from Moffat, H. at Sandia National Laboratories and several others. CANTERA can use a variety of mechanism formats and thermodynamic data representations, including those used by CHEMKIN and NASA. Refer to the above website or http://blue.caltech.edu/~dgg/dgg/Software.html for more information. GASEQ: A Chemical Equilibrium Program for Windows. GASEQ is a PCbased equilibrium program written by C. Morley that can solve several different types of problems including: composition at a defined temperature and pressure, adiabatic temperature and composition at constant pressure, composition at a defined temperature and at constant volume, adiabatic temperature and composition at constant volume, adiabatic compression and expansion, equilibrium constant calculations, and shock calculations. More information can found at the website http://www.arcl02.dsl.pipex.com/gseqmain.htm.

F. TEMPORAL KINETICS (STATIC AND FLOW REACTORS) SENKIN: “A Fortran Program for Predicting Homogeneous Gas Phase Chemical Kinetics With Sensitivity Analysis,” Lutz, A. E., Kee, R. J. and Miller, J.A. Sandia National Laboratories, Livermore, CA, Sandia Report SAND87-8248, 1987. A Fortran program (senkin.f) that solves the time evolution of a homogeneous reacting mixture. The program can solve adiabatic problems at constant pressure, at constant volume, or with a volume specified as a function of time. It can also solve constant pressure problems at constant temperature or with the temperature specified as a function of time. The code uses CHEMKIN-II/III for mechanism construction and DASAC software (dasac.f, M. Caracotsios and W.E. Stewart—U. Wisconsin) for solving the nonlinear ordinary differential equations. The program also performs a kinetic sensitivity analysis with respect to reaction rate constants. For the most recent versions, refer to the Reaction Design website http://www.reactiondesign.com/ lobby/open/index.html.

Appendix I: Programs for Combustion Kinetics

753

CONP: Kee, R. J., Rupley, F. and Miller, J. A. Sandia National Laboratories, Livermore, CA. A Fortran program (conp.f) that solves the timedependent kinetics of a homogeneous, constant pressure, adiabatic system. The program runs in conjunction with CHEMKIN and a stiff ordinary differential equation solver such as LSODE (lsode.f, Hindmarsh, A. C. “LSODE and LSODI, Two Initial Value Differential Equation Solvers,” ACM SIGNUM Newsletter, 15, 4, (1980)). The simplicity of the code is particularly valuable for those not familiar with CHEMKIN. HCT: (Hydrodynamics, Chemistry, and Transport)—“A General Computer Program for Calculating Time Dependent Phenomena Including OneDimensional Hydrodynamics, Transport, and Detailed Chemical Kinetics,” Lund, C. M. Lawrence Livermore National Laboratory, Report No. UCRL52504, Livermore, CA, 1978, Revised by Chase, L. 1989. A general-purpose model capable of modeling in detail one-dimensional time-dependent combustion of gases. The physical processes that are modeled are chemical reactions, thermal conduction, species diffusion, and hydrodynamics. Difference equations are solved by a generalized Newton’s iteration scheme. Examples of the types of problems that can be solved include homogeneous temporal kinetics, premixed freely propagating flames, and stirred reactors. LSENS: “A General Chemical Kinetics and Sensitivity Analysis Code for Homogeneous Gas-Phase Reactions,” Radhakrishnan, K. NYMA, Inc., Lewis Research Center Group, Brook Park, OH, NASA Reference Publication 1328, 1994. A Fortran computer code for homogeneous, gas-phase chemical kinetics computations with sensitivity analysis. A variety of chemical models can be considered: static system; steady, one-dimensional, inviscid flow; reaction behind an incident shock wave, including boundary layer correction; and perfectly stirred (highly backmixed) reactor. In addition, the chemical equilibrium state can be computed for the assigned states of temperature and pressure, enthalpy and pressure, temperature and volume, and internal energy and volume. Any reaction problem can be adiabatic, have an assigned heat transfer profile, or for static and flow problems, have an assigned temperature profile. For static problems, either the density is constant or the pressure-versustime profile is assigned. For flow problems, either the pressure or area can be assigned as a function of time or distance. CANTERA: Listed under Thermodynamic Equilibrium. COSILAB: Listed under Premixed Flames.

G. STIRRED REACTORS PSR: “A Fortran Program for Modeling Well-Stirred Reactors,” Glarborg, P., Kee, R. J., Grcar, J. F. and Miller, J. A. Sandia National Laboratories, Livermore, CA, Sandia Report SAND86-8209, 1986. PSR is a Fortran computer program (psr.f) that predicts the steady-state temperature and species

754

Combustion

composition in a perfectly stirred reactor. Input parameter include the reactor volume, residence time or mass flow rate, pressure, heat loss or temperature, and the incoming mixture composition and temperature. The equations are a system of nonlinear algebraic equations that are solved by using a hybrid Newton/time-integration method. The corresponding transient equations are solved because they do not require as good an initial estimate as the algebraic equations. The program runs in conjunction with the CHEMKIN-II/III package. In addition, first-order sensitivity coefficients of the mass fractions and temperature with respect to rate constants are calculated. SURPSR (surpsr.f, Moffat, M. E., Glarborg, P., Kee, R. J., Grcar, J. F. and Miller, J. A. Sandia National Laboratories) is a version of PSR that incorporates surface reactions via SURFACE CHEMKIN. For the most recent versions, refer to the Reaction Design website http://www.reactiondesign.com/lobby/open/index.html. LENS: Listed under Temporal Kinetics Calculations. CANTERA: Listed under Thermodynamic Equilibrium. COSILAB: Listed under Premixed Flames.

H. SHOCK TUBES SHOCK: “A General-Purpose Computer Program for Predicting Kinetic Behavior Behind Incident and Reflected Shocks,” Mitchell, R. E. and Kee, R. J. Sandia National Laboratories, Livermore, CA, Sandia Report SAND82-8205, 1982, reprinted 1990. A Fortran computer code (shock.f) for predicting the chemical changes that occur after the shock heating of reactive gas mixtures. Both incident and reflected shock waves can be treated with allowances for real gas behavior, boundary layer effects, and finite-rate chemistry. The program runs in conjunction with CHEMKIN as a preprocessor of the gas-phase mechanism and LSODE (lsode.f, Hindmarsh, A. C. “LSODE and LSODI, Two Initial Value Differential Equation Solvers,” ACM SIGNUM Newsletter 15, 4, 1980) to solve the stiff ordinary differential equations. LENS: Listed under Temporal Kinetics Calculations.

I. PREMIXED FLAMES PREMIX: “A Fortran Program for Modeling Steady Laminar One-Dimensional Premixed Flames,” Kee, R. J., Grcar, J. F., Smooke, M. D. and Miller, J. A. Sandia National Laboratories, Livermore, CA, Sandia Report SAND85-8240, 1985. A Fortran computer program that computes species and temperature profiles in steady-state burner-stabilized and freely propagating premixed laminar flames with detailed elementary chemistry and molecular transport. For the burner-stabilized problem, the temperature profile can either be specified or calculated from the energy conservation equation. For freely propagating flames,

Appendix I: Programs for Combustion Kinetics

755

flame speeds are calculated as an eigenvalue. Solution of the two-point boundary value problem is obtained by finite difference discretation and the Newton method (twopnt.f, Grcar, J. F., Sandia National Laboratories). Sensitivity analysis can also be performed. For the most recent versions, refer to the Reaction Design website http://www.reactiondesign.com/lobby/open/index.html. COSILAB: Combustion Simulation Software is a set of commercial software tools for simulating a variety of laminar flames including unstrained, premixed freely propagating flames, unstrained, premixed burner-stabilized flames, strained premixed flames, strained diffusion flames, strained partially premixed flames cylindrically and spherical symmetrical flames. The code can simulate transient spherically expanding and converging flames, droplets and streams of droplets in flames, sprays, tubular flames, combustion and/or evaporation of single spherical drops of liquid fuel, reactions in plug flow and perfectly stirred reactors, and problems of reactive boundary layers, such as open or enclosed jet flames, or flames in a wall boundary layer. The codes were developed from RUN-1DL, described below, and are now maintained and distributed by SoftPredict. Refer to the website http://www.softpredict.com/cms/ softpredict-home.html for more information. RUN-1DL—The Cambridge Universal Laminar Flamelet Computer Code: B. Rogg, in Reduced Kinetic Mechanisms for Applications in Combustion Systems, Appendix C, Peters, N. and Rogg, B. eds., Springer (Lecture Notes in Physics m15), Berlin/Heidelberg/New York, 1993. A Fortran computer code for the simulation of steady, laminar, one-dimensional and quasi one-dimensional, chemically reacting flows such as (i) unstrained, premixed freely propagating and burner-stabilized flames, (ii) strained, premixed flames, diffusion flames, and partially premixed diffusion flames, (iii) linearly, cylindrically, and spherically symmetrical flames, (iv) tubular flames, and (v) two-phase flames involving single droplets and sprays. The code employs detailed multicomponent models of chemistry, thermodynamics, and molecular transport, but simpler models can also be implemented. For example, the code accepts various chemistry models including detailed mechanisms of elementary reactions, systematically reduced kinetic mechanisms, one-step global finite-rate reactions, and the flame-sheet model (diffusion flames). Also implemented are radiation models, viz., a simple model based on the optically thin limit and a detailed model valid for the entire optical range. User-specified transport equations can also be provided to solve the equations for soot volume fraction and number density or the equations for a particle phase in particle-laden flames. For more information, see the description under COSILAB. FlameMaster v3.3: A C Computer Program for 0D Combustion and 1D Laminar Flame Calculations. FlameMaster was developed by H. Pitsch. The code includes homogeneous reactor or plug flow reactors, steady counter-flow diffusion flames with potential flow or plug flow boundary conditions, freely propagating premixed flames, and the steady and unsteady flamelet equations. More information can be obtained from http://www.stanford.edu/group/pitsch/Downloads.htm.

756

Combustion

HCT: Listed under Temporal Kinetics Calculations. CANTERA: Listed under Thermodynamic Equilibrium.

J. DIFFUSION FLAMES OPPDIF: “A Fortran Program for Computing Opposed-Flow Diffusion Flames,” Lutz, A. E., Kee, R. J. and Grcar, J. F. Sandia National Laboratories, Livermore, CA. A Fortran computer program for solving the one-dimensional axisymmetric diffusion flame between two opposing nozzles. OPPDIF solves for the temperature, species, normal and radial velocities, and the radial pressure gradient. The program uses CHEMKIN-II and the TWOPNT software package (Grcar, J. F. Sandia National Laboratories) to solve the two-point boundary value problem. For the most recent versions, refer to the Reaction Design website http://www.reactiondesign.com/lobby/open/index.html. CANTERA: Listed under Thermodynamic Equilibrium. COSILAB: Listed under Premixed Flames. FlameMaster v3.3: Listed under Premixed Flames.

K. BOUNDARY LAYER FLOW CRESLAF (Chemically Reacting Shear Layer Flow): “A Fortran Program for Modeling Laminar, Chemically Reacting, Boundary Layer Flow in Cylindrical or Planar Channels,” Coltrin, M. E., Moffat, H. K., Kee, R. J. and Rupley, F. M. Sandia National Laboratories, Livermore, CA, 1991 (see also Coltrin, M. E., Kee, R. J. and Miller, J. A. J. Electrochem. Soc. 133, 1206, 1986). CRESLAF is a Fortran computer program that predicts the species, temperature, and velocity profiles in two-dimensional (planar or axisymmetric) channels. The model uses the boundary layer approximations for the fluid flow equations, coupled to gas-phase and surface species continuity equations. The program runs in conjunction with CHEMKIN preprocessors (CHEMKIN, SURFACE CHEMKIN, and TRANFIT) for the gas-phase and surface chemical reaction mechanisms and transport properties. The finite difference representation of the defining equations forms a set of differential algebraic equations that are solved using the computer program DASSL (dassal.f, L.R. Petzold, Sandia National Laboratories Report, SAND 82-8637, 1982). COSILAB: Listed under Premixed Flames.

L. DETONATIONS CEA and CEC: Listed under Thermodynamic Equilibrium.

M. MODEL ANALYSIS AND MECHANISM REDUCTION CSP: “Using CSP to Understand Complex Chemical Kinetics,” Lam, S. H. Comb. Sci. Technol. 89, 375–404, 1993; “The CSP Method for Simplifying

Appendix I: Programs for Combustion Kinetics

757

Kinetics,” Lam, S. H. and Goussis, D. A. Int. J. Chem. Kinet. 26, 461–486, 1994 (www.Princeton.EDU/~lam/CSPCST.html). Computational Singular Perturbation is a systematic mathematical procedure to do boundary layer type singular perturbation analysis on massively complex chemical kinetic problems. It is a programable algorithm using CHEMKIN that generates not only the numerical solution, but can be used to obtain physical insights of the underlying kinetics from inspection of the numerical CSP data. The types of questions that may be addressed include (1) How to reduce the size of a chemical mechanism, (2) What are the rate controlling reactions, and (3) Which species can be approximated by algebraic relationships because of steady-state or partial equilibrium conditions? AIM: “An Improved Computational Method for Sensitivity Analysis: Green’s Function Method with AIM,” Kramer, M. A., Calo, J. M. and Rabitz, H. Appl. Math. Model. 5, 432, 1981. Program for performing sensitivity analysis of temporal chemical kinetic problems. The program allows for the calculation of first and second order sensitivity coefficients as well as for Green’s function coefficients. A detailed description of the usage and interpretation of these gradients is given by “Some Interpretive Aspects of Elementary Sensitivity Gradients in Combustion Kinetics Modeling,” R.A. Yetter, F.L. Dryer, and H. Rabitz, Comb. Flame 59, 107–133, 1985. The program is used in combination with other computer codes such as CONP. KINALC: Turanyi, T. Comput. Chem. 14, 253, 1990. Central Research Institute for Chemistry H-1525, Budapest, PO Box 17, Hungary (1995). KINALC is a postprocessor Fortran computer program (kinalc.f) for CHEMKINbased simulation programs including SENKIN, PSR, SHOCK, PREMIX, and EQUIL. The program performs three types of analysis: processing sensitivity analysis results, extracting information from reaction rates and stoichiometry, and providing kinetic information about species. Processing of sensitivity information includes calculating the sensitivity of objective functions formed from concentrations of several species. In addition, a principal component analysis of the sensitivity matrix can be performed. This eigenvector–eigenvalue analysis groups the parameters on the basis of their effect on the model output, e.g., it will show which parameters have to be changed simultaneously for a maximum change of the concentration of one or several species. KINALC carries out rateof-production analysis and gives a summary of important reactions. The matrix of normed reaction rate contributions can be considered as the sensitivity of reaction rates, and the principal component analysis of this matrix can be used for mechanism reduction. KINALC can estimate the instantaneous error of assumed steady-state species, and thus guide the selection of such species (Turanyi, T., Tomlin, A. and Pilling, M. J. J. Phys. Chem. 97, 163, 1993). KINALC accepts mechanisms with irreversible reactions only. MECHMOD is a Fortran computer program (mechmod.f) that transforms a mechanism with reversible reactions to one with pairs of irreversible reactions. Information on KINALC can be obtained from the following two websites: http://www.chem.leeds.ac.uk/Combustion/ Combustion.html and http://garfield.chem.elte.hu/Combustion/Combustion.html.

758

Combustion

XSENKPLOT—An Interactive, Graphics Postprocessor for Numerical Simulations of Chemical Kinetics: Burgess, D. Reacting Flows Group, Process Measurements Division, NIST, Gaithersburg, MD. An interactive, graphics postprocessor for numerical simulations of chemical kinetics calculations. This graphics postprocessor allows one to rapidly sort through and display species and reaction information generated in numerical simulations. Such interactive computations facilitate the development of a fundamental understanding of coupled chemically reacting systems by providing the ability to quickly probe the impact of process parameters and proposed mechanisms. The Fortran code is configured to be used in conjunction with the SENKIN and CHEMKIN computer codes.

Author Index

Abdel-Gayed, R. G., 235 Aders, W. K., 127, 131 Adigun, O., 66, 68 Ahn, C. C., 529 Ahn, C. H., 250 Ahn, T., 253 Aichlmayr, H. T., 250 Aksay, I. A., 252 Aksit, I. M., 471 Al-Alami, M. Z., 124, 125 Ali, A. N., 250 Allen, J. P., 435 Allen, M. G., 250 Altissimo, M., 250 Ang, S. S., 250 Antaki, P., 530 Anthony, D. B., 537 Antonsson, E. K., 250 Appleton, J. P., 431, 548 Aquino, M. I., 529 Armijo, A. D., 253 Aronowitz, D., 127 Arrhenius, S., 45 Ashurst, W. T., 233 Asthoiz, D. C., 135 Axelbaum, R. L., 315, 470, 471 Azarbarzin, A., 199 Babrauskas, V., 403, 404 Bachmeier, F., 425, 426, 427, 428, 429 Badzioch, S., 536 Baldwin, A. C., 113 Barenblatt, G. I., 155, 158, 382 Barnard, J. A., 130 Barnett, C. M., 399, 400 Barrett, R. E., 454 Baulch, D. L., 85 Beaucage, G., 250 Beck, D. D., 529

Beer, J. M., 547 Belles, F. E., 302, 305, 395, 396, 399, 400 Benjamin, A., 250 Benning, M. A., 524, 526 Benson, S. W., 43, 45, 48, 108, 110, 113, 131, 480, 482 Beretta, F., 474 Berkau, E. E., 431 Beskok, A., 251 Besser, R. S., 251 Bilger, R. W., 218 Bird, R. B., 155, 296 Birringer, R., 529 Bittner, J. D., 133, 475, 480 Blacet, F. R., 415 Blackshear, P. L. Jr., 339 Blanc, M. V., 398, 399, 401 Blasius, H., 367, 368 Blazowski, W. S., 460 Blumenberg, B., 123, 124, 135 Bodenstein, M., 53 Boedecker, L. R., 318 Boettner, J. C., 117, 239, 248 Bonczyk, P. A., 484 Bonini, J., 466 Borer, K., 253 Borghese, A., 474 Borghi, R., 235, 244, 249 Borovenskaya, I. P., 495, 514, 515, 519 Botha, J. P., 183 Bowden, F. P., 306 Bowman, C. T., 127, 328, 409, 417, 420, 421, 422, 424, 425, 429, 430, 433, 435, 437, 438, 439, 440, 441, 456 Bozzelli, J. W., 108 Brabbs, T. A., 115 Bracco, F. V., 436 Bradley, D., 214, 234, 235 759

760

Bradley, J. N., 55, 84, 147, 151, 201, 203, 207 Branch, M. C., 437 Brandt, O., 547 Bray, K. N. C., 218 Brewer, L., 497, 498, 505 Brezinsky, K., 118, 121, 122, 127, 129, 132, 133, 134, 135, 136, 137, 139, 405, 406, 466, 518 Broida, H. P., 448 Brokaw, R. S., 91, 93, 115, 189 Buckmaster, J. D., 155, 199 Budach, K. A., 124, 125 Bulfalini, M., 415, 416, 417 Bulian, C. J., 529 Bull, D. C., 300 Burke, E. J., 127, 134, 155 Burke, S. P., 323, 325 Butler, D. M., 64 Büki, A., 69 Cadou, C. P., 253 Calcote, H. F., 187, 399, 400, 460, 480 Caldwell, F. R., 182 Calo, J. M., 541 Cambel, A. B., 250 Cao, C. S., 251 Caracotsios, M., 66, 68 Carr, R. E., 353, 363, 364 Cashdollar, K. L., 401 Cathonnet, M., 113, 114, 117, 123, 239, 248 Cernansky, N. P., 435 Chaalane, A., 250 Chaos, M., 127, 128 Chapman, D. L., 265, 271 Chapman, S., 485 Charpenay, S., 538 Chase, M.W. Jr., 496, 497, 498 Chen, X. P., 250 Chigier, N., 253 Chiu, H. H., 364 Choi, H., 470 Choi, J. W., 250 Chou, C. P., 66, 68 Chou, S. K., 250 Choudbury, P. R., 250 Chung, S. H., 315

Author Index

Cicerone, R. J., 490, 491 Clark, T. P., 460, 462, 480, 482 Clary, D. W., 127, 480 Clavin, P., 232, 243 Clingman, W. H. Jr., 189 Cochran, T. H., 471 Cohen, R. B., 250 Cole, J. A., 127, 480 Colket, M. B., 128, 180 Coltrin, M. E., 66, 68, 169 Contic, R. S., 401 Conway, J. B., 497, 498, 505 Corley, T. L., 457 Courant, R. W., 434 Coward, H. F., 192, 199 Crittenden, B. D., 480 Crutzen, P. J., 487 Cukier, R. I., 64, 69 Cullis, C. F., 442, 443, 446 Cullis, H. F., 458, 460 Cunningham, A. C., 326, 328, 329 Curtiss, C. F., 155, 296 Dabora, E. K., 307 Dacol, D., 63, 64, 65 Dagaut, P., 113, 114, 117, 239, 248 Dahm, W. J. A., 250 Dainton, F. S., 54, 82, 417 Damkohler, G., 231 Darrieus, G., 227 Davies, C.A., 496, 497, 498 Davies, J.R. Jr., 496, 497, 498 Davis, K. A., 518, 519 Davis, S. C., 135, 138 Davis, W. C., 264 De Grandis, F., 250 Dean, A. M., 108 Denny, V. E., 360, 361, 362 Deshmukh, S. R., 251 deSoete, C. C., 433 deSoete, G. G., 457 Deutschmann, O., 253 Di Fabrizio, E., 250 Dibble, R. W., 66 DiBiaso, H. H., 250 Diehl, L. A., 434 DiLorenzo, A., 474 Dixon-Lewis, G., 66, 68, 169

Author Index

Do Conto, T., 250 Dobbins, R. A., 466, 474, 477 Dobbs, G. M., 318 Dodge, M. C., 64 Dorfman, R. C., 312, 313, 316, 317, 460, 469, 477 Dormal, M., 299 Dryer, F. L., 50, 58, 61, 65, 83, 87, 89, 90, 91, 94, 95, 113, 114, 115, 118, 119, 121, 122, 123, 125, 126, 127, 128, 129, 132, 152, 185, 186, 211, 212, 213, 252, 402, 436, 462 Du, J., 315 Dunn, B., 253 Dunn-Rankin, D., 250, 251, 253 Durant, J., 135 Döring, W., 265, 293 D’Alessio, A., 474, 477 Eastwood, C., 253 Eberius, K. H., 425, 426, 427, 428, 429 Eberstein, I. J., 7, 403, 407 Eckert, J., 529 Edelman, R. B., 241 Edwards, D. H., 302 Eklund, T., 212 Ekmann, J. M., 456, 457 El-Leathy, A. M., 547 Elsworth, J. E., 300 Emdee, J. L., 132, 133, 135, 136, 139 Emmons, H. W., 368 Engel, C. D., 300 English, B. A., 250 Epstein, A. H., 250 Esashi, M., 250 Essenhigh, R. H., 460 Esteve, D., 250 Evans, G. H., 66, 68 Faeth, G. M., 547 Fan, R. L., 250 Feng, C. C., 212, 222 Fenimore, C. P., 423, 428, 433, 547 Fernandez-Pello, A. C., 213, 250, 251 Fetting, F., 250 Fickett, W., 264 Fields, J. P., 241 Fineman, S., 373

761

Flagan, R. C., 431, 529 Fletcher, T. H., 538 Flock, E. S., 182 Flowers, W. L., 328 Fontijn, A., 50 Fortuin, C. M., 64, 69 Foulatier, R., 239, 248 Frank, P., 133 Frank-Kamenetskii, D. A., 384, 389, 520 Frenklach, M., 113, 114, 115, 117, 127, 480 Friedman, R., 155, 262 Fristrom, R. M., 117, 184 Frolov, Y., 529 Frost, E. E., 246 Fu, K., 250 Fu, Z., 529 Fukui, T., 250 Fulrip, D.J., 496, 497, 498 Gaillard, F., 239, 248 Gajdeczko, B. F., 252 Galant, S., 431 Gardiner, W. C. Jr., 83, 127, 480 Garo, A., 547 Gatica, J. E., 514 Gaydon, A. G., 311, 312 Gemci, T., 253 Gerardino, A., 250 Gershenzen, Y. M., 447 Gerstein, M., 182 Gibbs, G. J., 187 Gilmer, R. B., 399, 400 Glarborg, P., 66, 68, 240 Glassman, I., 7, 10, 21, 22, 50, 61, 95, 119, 121, 127, 128, 130, 132, 133, 134, 135, 136, 137, 139, 211, 212, 213, 222, 292, 293, 402, 403, 405, 406, 407, 458, 460, 461, 462, 464, 465, 466, 467, 468, 469, 472, 473, 474, 475, 476, 479, 480, 483, 484, 495, 496, 497, 498, 499, 509, 518, 524, 530 Godsave, G. A. E., 353, 358, 363 Golden, D. M., 51, 87, 89, 95, 113, 118, 122 Goldsmith, M., 353, 363 Gomez, A., 253, 464, 466, 474 Gordon, A. S., 93, 480

762

Gordon, S., 7, 21, 22, 286, 496 Gould, R. K., 518 Goussis, D. A., 63 Grant, D. M., 538 Graziani, F., 250 Grcar, J. F., 66, 68, 240 Great, J. F., 169 Grebe, J., 126 Gregory, C. A. Jr., 399, 400 Gregory, O. J., 529 Gromov, A. A., 529 Gu, Y., 250 Guerrero-Arias, I., 253 Guest, P. G., 398, 399, 401 Guillard, F., 117 Guirao, C., 300 Gutman, D., 108 Haggard, J. B., 471 Hainer, R., 391 Hamblen, D. G., 538 Hansel, J., 212 Hanson-Parr, D., 529 Harding, L. B., 86 Hargraves, K. J. A., 434 Harris, S. J., 466, 469, 475, 476 Harsha, P. T., 241 Harst, P. L., 450 Harvey, R., 434 Hawksley, P. G. W., 536 Hawthorne, W. R., 316, 330, 331 Hayhurst, A. N., 428, 429 Haynes, B. S., 458, 460, 474, 475, 476, 478 Haynes, D. S., 427, 433, 434 Hazard, H. R., 434 Hecht, T. A., 64 Held, T. J., 127, 129 Hertzberg, M., 401 Herzler, J., 133 Hibbard, R. R., 460, 462 Higa, K., 529 Hindmarsh, A. C., 63, 240 Hinshelwood, C. N., 382 Hirano, T., 213 Hirschfelder, J. O., 155, 296 Hiskey, M. A., 250 Hlavacek, V., 514, 529

Author Index

Ho, P., 66, 68 Holladay, J. D., 251 Holt, J. B., 514 Holzer, J. C., 529 Homann, K. H., 475 Homann, R. H., 126 Hong, C. C., 250 Hori, K., 250 Hosokawa, R., 250 Hottel, H. C., 61, 185, 235, 316, 330, 331, 537 Houf, W. G., 66, 68 Howard, J. B., 133, 475, 480, 536, 537, 547 Hoyermann, K., 123, 124, 135 Huang, Y. M., 250 Hubbard, G. L., 360, 361, 362 Huff, V. N., 21 Hummell, J. D., 454 Hunt, B. G., 486 Hunter, J. B., 113, 114, 117 Hunziker, H. E., 123, 125 Hura, H. S., 461, 472, 473, 474, 479, 483, 484 Hurt, R. H., 541 Hwang, J. Y., 315 Ibberson, V. J., 130 Ilyin, A. P., 529 Irvin, K. J., 417 Iverach, D., 427, 433, 434 Jackson, G. S., 253 James, H., 239, 248 Jandes, H., 474, 476 Janson, S. W., 250 Jarosinsky, J., 199 Jefferson, T. H., 68 Jiang, K. C., 250 Jin, P., 250 Johnson, C., 529 Johnson, G. M., 435, 443 Johnson, W. L., 529 Johnston, H., 487, 489 Jones, A. J., 302 Jones, E. O., 251 Jones, G. W., 547 Jones, O. W., 192, 199

Author Index

Jost, W., 394, 399 Jouguet, E., 265, 275 Ju, Y., 253 Just, T., 133, 425, 426, 427, 428, 429 Kafafi, S. A., 479, 480 Kailasanath, K., 199, 292 Kang, K. T., 315 Kanury, A. M., 352, 399 Karch, J., 529 Karniadakis, G. E., 251 Kaufman, F., 51 Kazakov, A., 127, 128 Kee, R. J., 66, 68, 24, 124, 169, 240, 437 Kennedy, I., 461, 469, 474, 476 Kent, J. H., 474, 476, 478 Kern, R. D., 480, 482 Kerr, T. T., 529 Kerrebrock, J. L., 250 Kerstein, A. R., 233, 538 Kesten, A. S., 436 Ketsdever, A. D., 250 Khudyakov, L., 331 Kiefer, J. H., 124, 125 Kim, B., 542 Kim, C. H., 547 Kim, S., 250 Kimura, K., 533 King, M. K., 530, 531 Kipping, F. B., 392 Kipping, F. S., 392 Kirov, N. Y., 427, 433, 434 Kistiakowsky, G. B., 296 Kittelson, D. B., 250 Klein, M. T., 140, 141 Klimov, A. M., 227, 229 Kneppe, H., 123, 125 Knipe, R., 93 Knobloch, A. J., 250 Knystautas, R., 300 Kobayashi, H., 536 Kopselis, S. A., 124, 125 Kramer, M., 63, 64, 65 Kruger, C. H. Jr., 48, 51 Kwon, S., 250 Kydd, J. P., 296 Kyritsis, D. C., 253

763

Lagrange, D., 250 Lahaye, J., 547 Laidler, K. J., 43, 46 Lam, S. H., 63, 69, 212, 222 Landau, L., 51, 227 Larangot, B., 250 Larson, C. W., 95, 118, 122 Larson, R. S., 66, 68 Law, A. V., 360 Law, C. K., 69, 359, 360, 361, 362, 363, 470, 471, 524 Le Chatelier, H. L., 153, 159 Le, N. B., 524, 526 Leach, T. T., 253 Leal, E. M., 250, 251 Leason, D. B., 190 Lee, C. H., 250 Lee, D. H., 250 Lee, J., 250 Lee, J. H., 263, 264, 265, 297, 300, 301 Lee, K. B., 547 Lee, S. B., 529 Lehn, H., 529 Lei, Y., 250 Leighton, P. A., 412, 413, 415 Leonard, R. A., 427 Levine, O., 182 Levinzon, D., 250 Levy, A., 432, 433, 434, 442, 446, 451, 457 Lewis, B., 83, 85, 87, 92, 154, 197, 200, 204, 205, 286, 303, 322, 325, 398, 399, 401 Lewis, D. H., 250 Li, C., 292 Li, G., 69 Li, J., 127, 128 Li, Y., 250 Libby, P. A., 218 Libouton, J. C., 299 Librovich, V. B., 155, 158, 382 Liepmann, D., 250 Lin, C. Y., 134 Lin, M. C., 134 Linan, A., 157, 158, 330, 331 Lindemann, F. A., 45, 57 Lindstedt, R. P., 135 Littman, M., 474

764

Litzinger, T. A., 113, 114, 117, 121, 135, 136, 137, 139 Liu, T. M., 364 London, A. P., 250 Long, R., 480 Longwell, J. P., 235, 246, 480 Lorell, J., 353, 363, 364 Loryan, V. E., 519 Lovell, A. B., 132, 133 Lu, T., 69 Ludford, G. S. S., 155 Lukachko, S., 250 Lung, F. K.-K., 235 Lutz, A. E., 66, 68 Lyon, R. K., 437 Ma, L., 542 Maas, U., 69 MacKiven, R., 212 Mahnen, G., 123 Makhviladze, G. M., 155, 158, 382 Makino, A., 533 Mallard, E., 153, 159 Mallard, W. G., 312, 313, 316, 317, 460, 469, 477 Manheimer-Timnat, Y., 462, 464, 471, 484 Manzoni, G., 250 Mao, C. P., 213 Marble, F. E., 245, 246 Markstein, G. H., 227, 228 Martenay, P. I., 424, 425 Martin, G. B., 431 Martinez, F. C., 250 Maruta, K., 250, 253 Marvin, C. S. Jr., 182 Masel, R. I., 253 Masi, S., 474 Mass, U., 66, 69 Matthews, C. J., 443 Mayor, R., 250 McBride, B. J., 7, 21, 22, 496 McBride, B. V., 286 McDonald, G. E., 460, 462, 480, 482 McDonald, R.A., 496, 497, 498 McKinnon, J. T., 135 McLellan, C. L., 66, 68 Meeks, E., 66, 68 Megaridis, C. M., 484

Author Index

Mehra, A., 250 Meissner, H. P., 537 Melius, C. F., 480 Mellor, A. M., 427 Merryman, E. L., 434, 442, 446, 451 Merzhanov, A. G., 495, 514, 515, 519 Metcalfe, E., 300 Micci, M. M., 250 Michael, J. V., 86 Mijit, K., 250 Mikhelson, V. A., 157, 159 Mikolaitis, D., 199 Milius, D. L., 252 Miller, J. A., 24, 66, 68, 124, 169, 240, 312, 313, 316, 317, 417, 422, 429, 437, 438, 439, 440, 441, 460, 469, 477, 480 Miller, S. F., 66, 68 Miller, W. J., 460, 518 Milliken, R. C., 462, 465 Mills, A. F., 360, 361, 362 Minkoff, G. J., 104 Miotti, P., 250 Mitchell, R. E., 66, 68, 124, 542 Miyaska, K., 250 Moffat, H. K., 66, 68 Molina, M. I., 489, 490 Moore, J. T., 503, 516 Morcomb, J. T., 457 Morell, J. E., 21 Morgan, A. G., 61 Moss, J. B., 471 Mulcahy, M. F. R., 435, 442, 443, 446 Muller, C. H., 448 Mullins, B. P., 394 Mullins, H. P., 380 Mulvihill, J. N., 431 Murakami, T., 250 Murugesan, S., 250 Myerson, A. L., 436, 450 Naegeli, D. W., 50, 95, 128 Nagle, J., 547 Nakajima, N., 250 Naltandyan, A. B., 447 Namikiri, T., 533 Nau, D. L., 250 Neoh, K. G., 547 Nerheim, N. M., 185, 235 Nesby, J. R., 480

765

Author Index

Ni, J., 250 Nicholson, H. M., 241 Niksa, S., 538 Nishida, O., 467 Noe, K., 199 Norris, R. G. W., 130 Norton, A. G., 251 Norton, D. G., 253 Norton, T. S., 127, 128 Ochoa, F., 253 Olson, D. B., 83, 518 Ondruschka, A., 141 Oppenheim, A. K., 263 Oran, E. S., 292 Ouyang, X., 251 Pagni, P. J., 213 Palmer, H. B., 458, 460 Palmer, K. N., 307 Palo, D. R., 251 Pamedemukkala, K. M., 480 Papas, P., 405, 406, 498, 499, 509 Papke, M., 250 Park, C., 548 Park, D. E., 250 Parker, W. G., 460 Parr, T. P., 529 Patnick, G., 199 Pease, R. D., 189 Pease, R. N., 154 Penner, S. S., 155, 353, 363 Perennes, F., 250 Perger, T., 69 Perry, R. A., 440 Peter, C. W., 536 Peters, I., 123 Petschek, A. G., 64, 69 Petzold, L. R., 66, 68 Phillips, L. F., 431 Phillips, P. F., 302 Pickett, L. M., 473, 474 Pisano, A. P., 250 Pivkina, A., 529 Plee, S. L., 427 Pola, A., 141 Pope, S. B., 69 Popenko, E. M., 529 Powling, J., 183

Prado, G., 547 Prakash, S., 253 Pranda, P., 529 Prandova, K., 529 Prewett, P. D., 250 Pugmire, R. J., 538 Puri, P., 529 Puri, R., 547 Puszynski, J. A., 529 Putnam, A. A., 250 Qiao, G., 250 Quilgars, A., 239, 248 Rabinovitz, M. J., 113, 114, 117 Rabitz, H., 58, 63, 64, 65, 69 Reid, W. T., 454 Remmler, M., 141 Reynolds, W. C., 21, 22, 66, 68 Ried, W. T., 442, 451 Roeder, C. H., 182 Rohsenow, W. M., 470 Ronney, P. D., 199, 234, 235, 252, 253 Roper, F. G., 326, 328, 329, 434 Rossi, C., 250 Roth, P., 547 Rouan, J. P., 239, 248 Rowland, F. S., 489, 490 Roychoudhury, S., 253 Rundel, R. D., 64 Rupley, F. M., 24, 66, 68, 169 Sachjan, G. A., 447 Saito, H., 250 Saito, K., 213 Sandler, S. I., 140, 141 Santoni, F., 250 Santoro, R. J., 127, 130, 312, 313, 316, 317, 460, 469, 474, 477, 547 Sarofim, A. F., 536, 547 Sarzynski, D., 108 Savage, L. D., 199 Sawyer, R. F., 21, 22, 293, 435 Schaibly, J. H., 64, 69 Schalla, R. L., 460, 462, 480, 482 Schefer, R. W., 543 Schelkin, K. L., 232, 233 Schneider, G. R., 185, 235 Schneider, S. J., 250

766

Schofield, K., 448 Schoonman, 529 Schug, K. P., 462, 464, 471, 484 Schumann, T. E. W., 323, 325 Scire, J. J. Jr., 127, 128 Scott, J. E. Jr., 7, 10 Scurlock, A. C., 244 Searcy, A. W., 505 Sedlackova, M., 141 Semedian, H. G., 474, 477 Semenov, N. N., 104, 106, 153, 382, 384, 394 Semerjian, H. G., 474, 477 Senturia, S. D., 250 Serio, M. A., 536, 538 Shaddix, C. R., 543 Shah, K., 251 Shannon, M. A., 253 Shchelkin, K. I., 300 Sheinson, R. S., 105 Sher, E., 250 Shiraishi, T., 250 Shuff, P. J., 300 Shuler, K. E., 64, 69 Shull, R. D., 529 Sichel, M., 307 Sidebotham, G., 460, 464, 467, 480, 482 Siebers, D. L., 440, 473, 474 Siegel, R. W., 529 Sievert, R., 123, 124, 135 Singh, H. J., 480 Sirignano, W. A., 253, 361, 362 Sitski, L., 253 Skevis, G., 135 Slagle, I. R., 108 Smith, C., 326, 328, 329 Smith, D., 434 Smith, I. W. M., 95, 541 Smith, M. Y., 435, 443 Smith, S. R., 480 Smooke, M. D., 66, 68, 69, 124, 169 Smyth, K. C., 312, 313, 316, 317, 460, 469, 477, 480, 547 Solomon, P. R., 535, 536, 538 Son, S. F., 250, 530 Spadaccini, C. M., 250 Spalding, D. B., 183., 198, 199, 201, 205, 246, 247, 339 Stanchi, S., 253

Author Index

Stein, S. E., 127, 479, 480 Steinberg, M., 448 Stewart, J., 140 Stewart, P., 95, 118, 122 Stewart, W. E., 66, 68 Stolarski, R. S., 64, 490, 491 Strauss, W. A., 307 Street, J. C., 460, 462, 464 Strehlow, R. A., 199, 286, 300 Strickland-Constable, R. F., 547 Sulmistras, A., 300 Sung, C.-J., 250, 292 Surgin, V. A., 529 Sutherland, J. W., 86 Suuberg, E. M., 536 Swett, C. C., 395, 396, 399, 400 Syvervd, A.N., 496, 497, 498 Takahashi, F., 460, 465, 466, 475 Takeda, K., 253 Tanaka, S., 250 Tanford, C., 154 Tang, X. S., 250 Taylor, F. R., 450 Taylor, G. W., 130 Teller, E., 51 Thiele, E. W., 541 Thomas, A., 460, 462, 464 Thring, M. W., 547 Tipper, C. F. H., 104 Todes, O. M., 394 Tokudome, S., 250 Tonkin, P. S., 307 Troe, J., 59, 135 Troshin, Y. K., 300 Tsang, W., 129 Tully, F. P., 123, 125 Turányi, T., 63, 64, 65, 69 Turns, S. R., 68, 503, 516 Ulyanova, P., 529 Urtiew, P., 263, 301 Vaccari, L., 250 van Tiggelen, P. J., 127, 299 Vance, I. M., 428, 429 Vandooren, J., 127 Vandsburger, U., 461, 469, 474, 476 Varatharajan, B., 69

767

Author Index

Venturini, J., 516 Vereshchagin, V. I., 529 Vican, J., 252 Vierbaum, R., 547 Vincenti, W. G., 48, 51 Vlachos, D. G., 251, 253 Volchko, S. J., 250 von Elbe, G., 83, 85, 87, 92, 154, 197, 200, 204, 205, 286, 303, 322, 325, 398, 399, 401 Von Gersum, S., 547 von Grosse, H., 497, 498, 505 von Karman, T., 155 von Neumann, J., 265, 293 Wagner, A. F., 86, 108 Wagner, H. G., 458, 460, 474, 475, 476, 478 Wahl, C., 133 Waitz, I. A., 250 Walters, S., 490, 491 Walther, D. C., 250, 251 Wang, C., 66, 68, 115 Wang, H., 113, 114, 117 Wang, Y., 251 Warnatz, J., 66, 68, 116, 121, 169 Watanabe, M., 250 Weddel, D. B., 330, 331 Weiner, A. M., 466, 469, 475, 476 Weiss, M. A., 235, 246 Welsch, L. M., 401 Wendt, H. R., 123, 125 Wendt, J. O. L., 456, 457 Westbrook, C. K., 83, 85, 89, 90, 91, 94, 95, 114, 115, 123, 125, 126, 127, 132, 152, 185, 186, 301 Westenberg, A. A., 117 Westmoreland, P. R., 480 Wetzel, E. D., 251, 253 Whitten, G., 487, 489 Wilhelm, R. H., 250 Williams, D. V., 443 Williams, F. A., 56, 69, 155, 157, 158, 218, 227, 229, 232, 241, 243, 246, 250, 264, 302, 326, 330, 331, 360, 367, 530 Williams, F. W., 105 Williams, G. C., 61, 185, 235 Williams, T. C., 543

Wilson, K., 529 Wise, H., 353, 363, 364 Wolfe, W., 403, 406 Wolfhard, H. G., 311, 312, 460 Wong, E. L., 182 Wright, W. J., 460, 462 Wu, B. C., 140, 141 Wu, C. H., 480, 482 Wu, K. T., 460 Xu, B., 253 Xu, F., 547 Yaccarino, P., 462, 464, 471, 477, 484 Yakhot, V., 233, 242 Yang, V., 529, 530 Yang, W., 250 Yeh, T. T., 474, 477 Yetter, R. A., 58, 65, 87, 89, 113, 252, 503, 516, 530 Yoffee, A. D., 306 Yong, J. N., 480 Yoon, E., 250 Young D. J., 66, 68 Young, P. D., 66, 68 Yu, C.-L., 115 Yu, Z.-Z., 538 Yuan, T., 480 Zabetakis, M. G., 380, 381 Zabrenski, J. S., 524, 526 Zachariah, M. R., 250, 529 Zamashchikov, V. V., 253 Zavyalov, S., 529 Zebatakis, K. S., 187, 188, 192, 194 Zeldovich, Y B., 382, 396, 398, 399, 421 Zeldovich, Y. H., 155, 158 Zeldovich, Y. N., 265, 293 Zellner, R., 50, 95 Zeppieri, S., 140 Zhang, H.-Y., 135 Zhang, J., 484 Zhang, K. L., 250 Zhang, X., 250 Zhao, Z., 127, 128 Zhou, Z. Y., 250 Zimmermann, G., 141 Zukoski, E. E., 245, 246

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Subject Index

Abstraction selectivity from hydrocarbons, 120, 121 Acetylenes, 99 oxidation, 123 Activated complex, 48 Activation energy, 46 asymptotics, 158 Air pollution, 409 Alcohols, 100 oxidation, 127 rate constants, 659 Aldehydes, 101 decomposition, 69 oxidation, 110 rate constants, 659 Alkyl compounds, 99 Aromatic compounds, 99 oxidation, 129 Arrhenius rate expression, 45 Atmospheric nitrogen kinetics, 420 Autoignition temperature AIT, 380 Automotive knock, 77 turbulence, 214 B, see Transfer number B variable, 342, 349 Benzene, 99 oxidation, 130 Beta scission rule, 122 Biradical, 54 Blasius boundary layer approach, 367 Blowoff, 203 gradient of, 203, 207 Bluff-body stabilization, 241 theories, 246 Boltzmann factor, 45 Bond energies, 693 Boron combustion, 530

Boundary layer thickness, 334 Branched chain reaction, 54, 77 Bunsen burner design, 207 Buoyant effects in fuel jets, 326 Burke–Schumann diffusion flame development, 322 Burning of condensed phases, 346, 352 of plastics, 372 Butadiene, 134 C2 flame radiation, 151 Carbon char combustion, 495 Carbon disulfide (CS2) oxidation, 449 Carbonyl sulfide (COS) oxidation, 449 Catalytic ignition, 406 CH flame radiation, 151 Chain ignition, 382 Chain length, 82, 83 Chain reactions, 53 branching, 54 critical branching factor, 81, 104 propagating, 54 termination, 54, 55 Chapman–Jouquet point, 150, 265, 266 Chemical mechanism of soot formation, 478 Chemical potential, 10 Chemical thermodynamics, 1 ClO catalytic cycle on ozone, 489 Closed and open thermodynamic systems, 2 CO oxidation and explosion limit, 91 rate constants, 659 Coal combustion, 495 Combustion kinetic programs, 747 Combustion synthesis, 513 compounds, 515 Combustion wave, see Flame speed 769

770

Compression ignition, 401 Cool flames, 104 Conversion factors, 556 Critical sooting equivalence ratios, 463 Cyanogen–oxygen flame, 155 Cycloparaffins, 99 Cyclopentadienyl radicals, 134 “d2” law, 333, 358, 363 Damkohler number, 222, 229, 246 Deflagration, see Flame speed Degenerate branching, 106 Detonation, 261 cellular, 262, 297 dynamic parameters, 264, 301 effect of fuel molecular weight, 292 experimental results, 286 limits, 302 nongaseous media, 306 transition from deflagration, 261, 262 velocity, 264, 282 wave structure, 293 Diffusion flames, 261, 311 appearance, 312, 313, 314, 315, 316, 339, 340 buoyancy effects, 326 species distribution, 357 structure, 316 theoretical considerations, 318, 337, 347 turbulent fuel jets, 329 Diffusional kinetics, 520 Droplet flames, 331, 332, 346 clouds, 364 convective conditions, 365 flame position, 353 flame temperature, 353 mass burning rate, 352 with radiation, 372 refinements to theory, 359 Eddy viscosity, 232 Elemental sulfur oxidation, 451 Equilibrium constant(s), 8 concentration, 13, 14 constant pressure, 12, 15 Equilibrium of a chemical system, 8, 21

Subject Index

Equivalence ratio, 22, 23 Ethane oxidation, 118, 119 rate constants, 659 Ethanol, 100 oxidation, 128 Ethers, 103 Evaporation coefficient, 332, 359 Explosion, 75, 261, 379 Explosion limits, 83, 91, 98, 103 Explosive mixtures, 76 reaction, 43, 75 Extensive properties, 9 “Fall-off” range, 57, 58 Fick’s law, 319 First law of thermodynamics, 3 Flame properties, 169, 176 speed, laminar, 153 comprehensive theory, 168 data, 713 experimental results, 185, 187, 188, 189 Mallard–LeChatelier theory, 156 measurements, 176 Mikhelson theory, 157 pressure effect, 159, 160 Zeldovich, Frank-Kamenetskii– Semenov theory, 161 stabilization, 201 low velocity, 201 flash-back and blowoff, 203 high velocity, 240 stretch, 227 structure, 151, 168 suppressants, 191 temperature, 1, 7, 16, 28, 152 effect of pressure, 28, 29, 31 product composition, 30, 31 thickness, 157, 160 Flammability limits, 84, 192 data, 703 Forced ignition, 395 Formation of nitrogen oxides, 417 Fourier law of heat conduction, 336 Frank-Kamenetskii numbers, 222 Free energy, 8

771

Subject Index

Freons (halocarbons) and ozone, 491 Fuel-bound nitrogen reaction mechanisms, 431 Gaseous fuel jets, 311 appearance, 312 phenomenological analysis, 328 structure, 316 theoretical considerations, 318 turbulent, 329 Glassman’s criterion for vapor-phase combustion of metal, 496 Gordon–McBride program, 21, 22, 24 Grashof number, 367 H atom recombination, 44 H2—Br2 reaction, 54, 76 H2—Cl2 reaction, 77 H2—I2 reaction, 52, 77 H2—O2 reaction, 84 explosion limits, 84, 85 rate constants, 659 H2S oxidation, 444 Heat of vaporization, 333 Heat transfer coefficient, 334 Heat(s) of formation, 1, 4, 5, 6 metal oxides, 516 metal chlorides, 518, 519 metal nitrides, 519 Heat(s) of reaction, 1, 2, 3, 4 Heats of volatilization of metal oxides and nitrides, 507 HOx catalytic cycle on ozone, 486 Hugoniot curve, 149, 270 relation, 149 Hybrid rocketburning rate, 367 Hydrazine, 7, 406, 407 Hydrocarbon oxidation, 98 acetylene, 123 aliphatic, 117 aromatic, 129 general mechanism, 126 isomerization steps, 108 low temperature, 106 methane, 112 olefin, 123 paraffin, 120

Hydroperoxy radical, 86, 174 Hypergolicity, 403 Ignition, 338 chain, 382 thermal, 384 temperature, 340 data, 721 Induction intervals, 103 Intensive properties, 9 Karlovitz flame stretch factor, 227, 230 Ketones, 101 Kinetic rate parameters, 747 Klimov–Williams criterion, 229 Kolmogorov length (scale), 220 Law of heat summation, 4 Law of mass action, 44 Le Chatelier’s principle, 18, 28 Lewis number, 162 Liftoff of fuel jets, 331 Lindemann’s theory, 45 Longitudinal burning surface, 367 Mallard–Le Chatelier theory of flame speed, 156 Mass burning rate of nonvolatile fuels, 524 Mass diffusivity coefficient, 161, 318, 336 Metal combustion, 495 in high oxygen concentrations, 524 pressure effect on temperature, 505 temperatures, 498, 499, 509 Metal nitride properties, 497 Metal oxide properties, 497 vaporization-decomposition, 496 Methane flame velocities, 169, 175 Methane oxidation, 112, 113, 186, 187, 189 rate constants, 659 Mikhelson theory of flame speed, 157 Minimum ignition energy, 396, 400 Model analysis and mechanism reduction, 756

772

Momentum diffusivity, 162, 336 Monopropellants, 7, 21, 46, 406 Nagle and Strickland-Constable, 548 Negative temperature coefficients of reaction rate, 104 Newton-Raphson method, 22 Newton’s law of viscosity, 336 Nitric oxide reaction mechanisms, 420 rate constants, 677 Nitrogen dioxide reaction mechanisms, 433 rate constants, 677 Nitrogen oxides structures, 418 Nitrous oxide formation mechanisms, 435 NOx catalytic cycle on ozone, 487 NOx reduction, 436 RAPRENO process, 440 thermal DeNOx process, 439 NOx—SOx interactions, 456 Non-Arrhenius behavior, 51 Nusselt number, 334, 370, 371 Oil spills, 403 Olefins, 99 oxidation, 123 reactions, 126 Open thermodynamic systems, 2 Order of chemical reaction, 44 Organic acids, 102 Organic compound nomenclature, 99 Organic sulfur compounds oxidation, 451 Ozone decomposition, 70 rate constants, 684 Paraffins, 99 oxidation of, 120 Partial equilibrium assumption, 60 Partial pressure, 11 Partition function, 49 Peroxide, 103 Peroxide bonding, 110 Photochemical initiation, 76 Photochemical smog, 410 effect of NOx, 411 effect of SOx, 415 Photoyield, 76

Subject Index

Piling and Bedworth number, 404 Plug flow reactor, 236 Prandtl number, 162, 220 Preheat zone, 151 Prompt NO mechanisms, 423 Propane oxidation, 119 Pseudo first order, 57 Pyrophoricity, 403, 404, 405 Quenching distance, 200, 400 data, 743 Radiant heat flux, 335, 374 Ramjets, 32, 403 Rates of reaction, 43 abstraction, 120 chain branching, 54, 75 negative coefficient, 104 pressure effect, 61 recombination, 51 simultaneous interdependent, 52 steady-state assumption, 56, 60 theories, 47, 50 Reaction rate coefficient, 44 constants, 659 Reaction zone thickness, 157, 158, 173, 176 Reduction of nitrogen oxides, 417 Reference state, 2 Rockets, 5, 46 Standard state, 2 Stanjan, 22 Schlieren technique, 179 Schmidt number, 162, 220 Scramjet, 33 Self-propagating high-temperature synthesis (SHS), 514 Shadowgraph technique, 178, 179 Silane, 403, 404 Similarity parameters, 162 Simultaneous interdependent reactions, 52 Smog, 409 Soot characteristics, 458 Soot combustion, 495

773

Subject Index

Soot formation processes, 459 experimental measurement procedures, 460 influence of physical and chemical parameters, 482 Soot growth, 475 Soot inception temperature, 467 Sooting tendencies, 462 diffusion flames, 466 premixed flames, 464 SOx emissions, 441 Spark ignition energy, 396 data, 743 Spontaneous combustion, 392 ignition, 382 Stability limits of laminar flames, 191 Stagnant film burning, 365 Steady state assumption, 56, 60 Steric factor in kinetics, 47 Stirred reactors, 235 Stratospheric ozone, 485 Strong deflagration, 274 Strong detonation, 274 Sulfur compound structures, 444 oxidation products, 443 trioxide reactions, 454 rate constants, 685 Supersonic combustion, 33, 403 Surface oxidation, 495

Temporal kinetics, 752 in boundary layers, 756 in detonations, 756 in diffusion flame, 756 in flow reactors, 752 in premixed flames, 754 in shock tubes, 754 in static reactors, 752 in stirred reactors, 753 Thermal diffusivity, 336 Thermochemical data, 555, 557 parameters, 747 Thermodynamic equilibrium programs, 750 Tilted flames, 203 Transfer number B, 181, 339, 344, 346, 351 small B assumption, 372 values, 353 Transition state theory of reaction rate, 47 Transport parameters, 748 Triethyl aluminum (TEA), 403 Turbulent flame speed, 225, 231 Clavin, 232, 233 flames, 214 fuel jet, 329 reacting flow, 213, 216, 218 reaction rate, 216 Reynold’s number, 219

Taylor length (microscale), 220 Taylor microscale, 220 Thermal ignition, 384, 389 Thermite reaction, 514 Thermodynamic systems, 2 closed, 2 open, 2

Unsymmetrical dimethyl hydrazine, 403 Vinyl acetylene, 134 Viscosity, 336 Weak deflagration, 274 detonation, 274